(We're working in ZFC + $V=L$.) Let $\alpha$ be the least admissible $>\omega_1$. Then $\mathrm{Sat}_\alpha$ is not definable from parameters over $L_\alpha$.
Lemma: (Assuming ZFC + $V=L$.) Let $\kappa$ be a cardinal such that $\mathrm{cof}(\kappa)>\omega$. Let $M$ be a model of "$V=L$" with $\kappa\subseteq$ the wellfounded part of $M$ and $n<\omega$ and $x\in M$ be such that $M$ is the $\Sigma_n$-hull (over $M$) of elements in $\kappa\cup\{x\}$. Suppose there is no strictly decreasing $\omega$-sequence of $M$-ordinals which is $\Sigma_n$-definable from parameters in $M$. Then $M$ is wellfounded.
Proof: Suppose $M$ is illfounded and let $\left<\alpha_i\right>_{i<\omega}$ be strictly descending in $M$'s ordinals. We can pick a sequence $\left<\varphi_i,\beta_i\right>_{i<\omega}$ such that $\varphi_i$ is a $\Sigma_n$ formula which defines $\alpha_i$ from $(\beta_i,x)$. But the sequence $\left<\varphi_i,\beta_i\right>_{i<\omega}\in L_\kappa\subseteq M$, and therefore $\left<\alpha_i\right>_{i<\omega}$ is $\Sigma_n$-definable from the parameter $\left<\varphi_i,\beta_i\right>_{i<\omega}$ over $M$, contradiction.
Now fix $\alpha$ the least admissible $>\omega_1$. Then $L_\alpha$ is the $\Sigma_1$-hull of $\omega_1+1$. So the full $L_\alpha$-theory of parameters in $\omega_1+1$ is not definable from parameters over $L_\alpha$. Let $\lambda=\omega_1+1$. Let $T$ be the $\mathscr{L}_{\infty,\omega}\cap L_\alpha$-theory, using $\in$, $=$ and constant symbols $c$ for each $c\in\lambda$, asserting the conjunction of (i) KP + $V=L$, (ii) I am wellfounded through $\lambda$ (i.e. "$0,1,\ldots,\lambda$ are an initial segment of my ordinals"), (iii) there is no proper segment of me beyond $\lambda$ which satisfies KP,
(iv) I am the $\Sigma_1$-hull of $\lambda$, and (v) there is no infinite strictly decreasing sequence through my ordinals which is $\Sigma_1$-definable from parameters. Note that by the lemma, $L_\alpha$ is the unique model of $T$.
But now for each $\vec{\xi}\in(\omega_1+1)^{<\omega}$ and formula $\varphi$, the theory $T+\varphi(\vec{\xi})$ is satisfiable iff $L_\alpha\models\varphi(\vec{\xi})$. Therefore $L_\alpha$ cannot define satisfiability from parameters.
Taking this a bit further, we get:
Theorem: (Assume ZFC + $V=L$.) Let $\alpha$ be an infinite ordinal which is not a cardinal, such that $\mathrm{card}(\alpha)$ has cofinality $>\omega$. Then $\mathrm{Sat}_\alpha$ is definable from parameters over $L_\alpha$ iff $L_\alpha$ is closed under models witnessing its satisfiable formulas, i.e. iff \begin{equation} \mathrm{Sat}_\alpha(T)\iff L_\alpha\models
\text{"there is a model of }T\text{"}.\end{equation}
Proof Sketch: Suppose there is $T\in L_\alpha$ which is satisfiable but which has no model in $L_\alpha$. Like mentioned in the original question, $T$ has a model in $L_{\alpha^+}$. Therefore there is $\beta$ such that $\alpha\leq\beta<\alpha^+$
and $L_\beta$ projects to $\kappa=\mathrm{card}(\alpha)$ (i.e. there is $n<\omega$ and $x\in L_\beta$ such that $L_\beta$ is the $\Sigma_n$-hull of parameters in $\kappa\cup\{x\}$) and there is a model of $T$ in $L_\beta$. But since $T\in L_\alpha$, then arguing much as before, we can construct a theory $T'\in L_\alpha$ whose unique model is the least such $L_\beta$, which implies that $L_\alpha$ cannot define satisfiability, much as before. QED.
This leaves the uncountable non-cardinals $\alpha$ where $\mathrm{card}(\alpha)$ has countable cofinality, which the above obviously doesn't touch.
To move this off the unanswered queue, I'm turning Harry West's comment above into an answer. If he posts an answer of his own, I'll delete this one and accept it instead; meanwhile, I've made this CW to avoid reputation gain.
The answer is yes, at least if we're willing to accept rather strong large cardinal axioms as consistent with $\mathsf{ZFC}$: the principle CMI follows for example from the existence of a proper class of strongly compact cardinals. This is because every structure can be described up to isomorphism by a single $\mathcal{L}_{\mu,\mu}$-sentence for some sufficiently large $\mu$ (the successor of the cardinality of the structure is always enough, but sometimes we can do better).
So suppose $\mathcal{A}$ is a $\Sigma$-structure and $\kappa_0$ is a cardinal. Let $\theta$ be a strongly compact cardinal $>\vert\mathcal{A}\vert+\kappa_0$ and suppose $T$ is a theory of arbitrary cardinality, in an expansion $\Sigma'$ of $\Sigma$ by constant symbols, every size-$<\theta$ subtheory of which has a model with $\Sigma$-reduct $\mathcal{A}$. Let $\varphi$ be a $\mathcal{L}_{\mu,\mu}$-sentence characterizing $\mathcal{A}$ up to isomorphism, with $\mu<\theta$; this exists since $\vert\mathcal{A}\vert<\theta$. Now just apply the strong compactness of $\theta$ to the theory $T\cup\{\varphi\}$.
Best Answer
(Edit: This replaces an earlier attempt I had in the opposite direction, which was bogus, as pointed out by @HarryWest, and to which some comments below refer.)
(Edit 2: It now gets weak compactness as a consistency strength lower bound.)
(Edit 3: A part initially missing at the end of the Subcase 2.2 argument has been filled in.)
(Edit 4: 2 observations specifically on $(\kappa,\kappa^{++})$-compactness added at the end.)
Remark: It looks like @HarryWest has already answered the original question. Below I work more directly with $\mathcal{R}$-satisfiability and extract some more strength. I don't know to what extent Harry's method would also lead to the following.
Remark 2: The reader unfamiliar with the Dodd-Jensen core model $K^{\mathrm{DJ}}$, but familiar with $0^\sharp$, should replace "there is an inner model with a measurable cardinal" in the claim below with "$0^\sharp$ exists", and replace $K^{\mathrm{DJ}}$ with $L$ throughout; this results in a weaker result, but the proof is essentially identical, and it suffices for Corollaries 1 and 3.
Claim. Assume ZFC. Let $\kappa$ be an uncountable cardinal such that $\mathcal{R}$-satisfiability is $(\kappa,\kappa^+)$-compact. Then there is a weakly inaccessible cardinal $\mu\leq\kappa$, and either:
there is an inner model with a measurable cardinal, or
letting $K=K^{\mathrm{DJ}}$ be the Dodd-Jensen core model below a measurable, $K\models$"$\kappa$ is weakly compact", and if $2^\gamma\leq\kappa$ for all $\gamma<\kappa$ then $\kappa$ is weakly compact.
Corollary 1. ZFC + "There is an uncountable cardinal $\kappa$ such that $\mathcal{R}$-satisfiability is $(\kappa,\kappa^+)$-compact" is equiconsistent with ZFC + "There is a weakly compact cardinal".
Corollary 2. Assume ZFC + GCH + "there is no inner model with a measurable cardinal". Let $\kappa$ be an uncountable cardinal. Then $\mathcal{R}$-satisfiability is $(\kappa,\kappa^+)$-compact iff $\kappa$ is weakly compact.
Corollary 3. Assume ZFC + GCH + "$0^\sharp$ does not exist". Let $\kappa$ be an uncountable cardinal. Then $\mathcal{R}$-satisfiability is $(\kappa,\kappa^+)$-compact iff $\kappa$ is weakly compact.
(Note that the converse direction of Corollaries 2 and 3, i.e. if $\kappa$ is weakly compact then $\mathcal{R}$-satisfiability is $(\kappa,\kappa^+)$-compact, holds in general.)
Proof of Claim: Fix $\kappa$. Except for the proof that there is a weakly inaccessible cardinal $\leq\kappa$, we may assume there is no inner model with a measurable cardinal, so write $K=K^{\mathrm{DJ}}$. (As mentioned in Remark 2, the reader unfamiliar with $K$ but familiar with $0^\#$ should just assume $0^\#$ does not exist, in which case $K=L$.) I will formally assume this, but it is easy to drop this assumption and still get the weakly inaccessible $\mu\leq\kappa$.
The plan is to find a reasonable elementary substructure $\bar{\mathcal{H}}$ of some fragment of $V$ and an elementary embedding $j:\bar{\mathcal{H}}\to M$ with a critical point $\mu$, and use this to get the desired conclusions.
If there is $\gamma<\kappa$ such that $2^\gamma\geq\kappa$ then letting $\gamma$ be least such, fix a sequence $\left<A_\alpha\right>_{\alpha<\kappa}$ of pairwise distinct subsets of $\gamma$. Otherwise let $A_\alpha=\emptyset$ for all $\alpha<\kappa$. Let $\mathcal{H}=(\mathcal{H}_{\kappa^+},\vec{A},<^*)$, where $<^*$ is a wellorder of $\mathcal{H}_{\kappa^+}$ (the set of all sets hereditarily of cardinality $\leq\kappa$).
We will build a theory $T$, of size $\kappa$, such that every sub-theory of size ${<\kappa}$ is $\mathcal{R}$-realizable. Basically, we want $T$ to describe a model which contains a version of $\mathcal{H}$ as an element, together with the statement "$\kappa$ is not a cardinal". The theory $T$ will use primary constants $\dot{\vec{\alpha}}$ for certain sequences $\vec{\alpha}$ of ordinals ${<\kappa^+}$. The sequences $\vec{\alpha}$ used we call the relevant sequences, and which are considered relevant depends on the following cases. If (i) $\gamma^{\omega}\leq\kappa$ for all $\gamma<\kappa$ then there are only $\kappa$-many $\omega$-sequences $\vec{\alpha}\in{^\omega}\kappa$ such that $\vec{\alpha}$ is bounded in $\kappa$ (and of course if $\mathrm{cof}(\kappa)>\omega$, "bounded in $\kappa$" can be struck out), and in this case these sequences $\vec{\alpha}$ are the relevant ones. Suppose instead (ii) there is $\gamma<\kappa$ such that $\gamma^{\omega}>\kappa$. If (ii.1) $\kappa^{+K}<\kappa^+$ then the relevant sequences are just the finite tuples $\vec{\alpha}\in\kappa^{<\omega}$. Suppose instead (ii.2) $\kappa^{+K}=\kappa^+$. In this case we need to be a little more careful. Fix from now on an ordinal $\eta\in(\kappa,\kappa^+)$ of cofinality $\kappa$ and such that letting $\bar{\mathcal{H}}=\mathrm{Hull}^{\mathcal{H}}(\eta)$, we have $\eta=\bar{\mathcal{H}}\cap\eta$. Then the relevant sequences are the finite tuples $\vec{\alpha}\in\eta^{<\omega}$. (Also for the proof that there is a weakly inaccessible cardinal $\mu\leq\kappa$, i.e. without assuming $K$ exists, in case (ii), proceed as in (ii.1).)
Using the process in the answer here: https://mathoverflow.net/questions/394526/is-this-compactness-property-for-satisfiability-on-mathbbr-consistent, we then also augment the theory with some secondary constants, so as to allow us to talk about subsets of $\omega$ coded by reals, and to make arithmetic statements about those coded sets using (infinitely many) statements in $T$. We will skip the details of those extra constants, and just directly make arithmetic statements about the coded subsets of $\omega$.
For each relevant $\vec{\alpha}$, let $t_{\vec{\alpha}}$ be the full theory of the single parameter $\vec{\alpha}$ in the structure $\mathcal{H}$ (note this includes predicates for $\vec{A}$ and $<^*$). Add the following statements to $T$ (they mostly refer to theories coded by reals):
$\dot{\vec{\alpha}}$ codes a consistent complete theory $u_{\dot{\vec{\alpha}}}$ in the language of set theory augmented with symbols $\hat{\vec{\beta}},\hat{\mathcal{H}},\hat{f},\hat{\kappa},\hat{\xi}$,
$u_{\dot{\vec{\alpha}}}$ contains the formula "$\hat{\mathcal{H}}=(\mathcal{J},<',\vec{A}')$ is a structure with transitive universe $\hat{\mathcal{J}}$, $<'$ is a wellorder of $\mathcal{J}$, and $\hat{\kappa}$ is the largest cardinal of $\mathcal{J}$",
$u_{\dot{\vec{\alpha}}}$ contains the formula "$V=L(\hat{\mathcal{H}},\hat{f})$ and $\hat{\xi}<\hat{\kappa}$ and $\hat{f}:\hat{\xi}\to\hat{\kappa}$ is a surjection",
the model determined by $u_{\dot{\vec{\alpha}}}$ has standard $\omega$,
and for each formula $\varphi[\vec{\alpha}]\in t_{\vec{\alpha}}$, add the statement
to $T$. Moreover, if $\vec{\gamma}$ is also relevant and $\mathrm{rg}(\vec{\alpha})\subseteq\mathrm{rg}(\vec{\gamma})$ and in the $\omega$-sequence case, $\vec{\alpha}$ itself is easily computed from $\vec{\gamma}$ (say there is a recursive function $i:\omega\to\omega$ such that $\vec{\alpha}_n=\vec{\gamma}_{i(n)}$), then we add the formula
to $T$.
Now let $S\subseteq T$ be a sub-theory of size ${<\kappa}$. We find an $\mathcal{R}$-realization of $S$. Let $C$ be the set of relevant sequences $\vec{\alpha}$ used in $S$; so $C$ has size ${<\kappa}$, and $C\subseteq\mathcal{H}$. Let $H=\mathrm{Hull}^{\mathcal{H}}(C)$; that is, the structure whose universe is the set of all elements of $\mathcal{H}_{\kappa^+}$ definable over $\mathcal{H}$ from parameters in $C$ (using the predicates of $\mathcal{H}$), and with predicates being the restrictions of those of $\mathcal{H}$. So $H\preccurlyeq\mathcal{H}$. Let $\bar{H}$ be the transitive collapse of $H$. Let $\pi:\bar{H}\to H$ be the uncollapse map. Let $\bar{\kappa}=\pi^{-1}(\kappa)$ etc. By enlarging $S,C$ if necessary, we may assume that $\bar{\kappa}$ has cardinality ${\xi<\bar{\kappa}}$. So let $f:\xi\to\bar{\kappa}$ be a surjection. Let $N=L_\beta(\bar{H},f)$ for some ordinal $\beta>0$.
Now for each $\vec{\alpha}\in C$, let $u_{\vec{\alpha}}$ be the theory in $N$ of the parameters $\bar{\vec{\alpha}},\bar{H},f,\xi,\bar{\kappa}$, where bars denote preimage under $\pi$. Then note that by interpreting $\dot{\vec{\alpha}}$ as the real naturally coding $u_{\vec{\alpha}}$, we get an $\mathcal{R}$-realization of $S$.
By $(\kappa,\kappa^+)$-compactness, we can fix an $\mathcal{R}$-realization $\mathcal{R}^+$ of $T$. Let $u_{\vec{\alpha}}$ be the theory coded by $\dot{\vec{\alpha}}^{\mathcal{R}^+}$. Note that we can define a term model $M$, pointwise definable from constants $\widetilde{\vec{\alpha}}$ (for relevant sequences $\vec{\alpha}$ as before) and constants $\widetilde{\mathcal{H}},\widetilde{f},\widetilde{\xi},\widetilde{\kappa}$, and such that $u_{\vec{\alpha}}$ is just the theory in $M$ of $\widetilde{\vec{\alpha}}^M,\widetilde{\mathcal{H}}^M,\widetilde{f}^M,\widetilde{\xi}^M,\widetilde{\kappa}^M$ (the super-$M$ denotes the interpretation of a constant in $M$).
Let $\bar{\mathcal{H}}=\mathrm{Hull}^{\mathcal{H}}(C)$ where $C$ is the set of all relevant sequences (the hull is uncollapsed); note that $\bar{\mathcal{H}}$ is in fact transitive (so actually it's the same as the collapsed hull) so $\bar{\mathcal{H}}\preccurlyeq\mathcal{H}$ (here $\bar{\mathcal{H}}$ is a structure with predicates induced by those of $\mathcal{H}$) (and note that in case (ii.2), what we defined as $\bar{\mathcal{H}}$ earlier is the same as the model we have just now defined, and so in this case $\mathrm{OR}\cap\bar{\mathcal{H}}=\eta$). We have $\kappa+1\subseteq\bar{\mathcal{H}}$. (But $\kappa^+\not\subseteq\bar{\mathcal{H}}$, since $\bar{\mathcal{H}}$ has cardinality $\kappa$.) Note that there is an elementary embedding $j:\bar{\mathcal{H}}\to\widetilde{\mathcal{H}}^M$ given by setting $j(\vec{\alpha})=\widetilde{\vec{\alpha}}^M$ for each relevant $\vec{\alpha}$.
We have $j(\kappa)=\kappa^M$. Since $\kappa$ is a cardinal in $V$, but $\kappa^M$ is not a cardinal in $M$ (though it is of course a cardinal in $\widetilde{\mathcal{H}}^M$), $j$ has a critical point $\mu\leq\kappa$ (here $M$ might be illfounded). So $\mu$ is a regular cardinal in $\bar{\mathcal{H}}$, and hence also regular in $V$. Note that $\mu>\omega$, as $M$ has standard $\omega$, as this held for each sub-model coded by $u_{\vec{\alpha}}$. Note that $\mu$ is not a successor cardinal (if $\mu=\gamma^+$ then $$\gamma^+=\mu<j(\mu)=j(\gamma^+)=j(\gamma)^{+M}=\gamma^{+M},$$ so $\mu$ is collapsed in $M$, a contradiction). So $\mu$ is weakly inaccessible, in particular giving the existence of a weakly inaccessible $\leq\kappa$ (we didn't yet use $K$). So from now on we do assume we have $K$, i.e. there is no inner model with a measurable.
Case 1: $\gamma^{\omega}\leq\kappa$ for all $\gamma<\kappa$.
So the relvant sequences are the bounded-in-$\kappa$ $\omega$-sequences $\vec{\alpha}$. We consider two subcases.
Subcase 1.1: $\mu<\kappa$.
In this case there is an inner model with a measurable cardinal. For if not, then the Dodd-Jensen core model $K$ (below an inner model with a measurable cardinal) exists. So there is no transitive proper class $K'$ and non-trivial elementary $k:K\to K'$. And $K\models GCH$, and $\mathcal{P}(\mu)\cap K\subseteq\bar{\mathcal{H}}$.
Let $\delta$ be some "ordinal" of $M$ such that $\beta<\delta<j(\mu)$ for all $\beta<\mu$ (it doesn't matter whether $M$ is wellfounded; but we will use that $M$ has wellfounded $\omega$). Let $U$ be the $K$-ultrafilter derived from $j$ with seed $\delta$. Then $\mathrm{Ult}(K,U)$ is wellfounded. For suppose not, and let $\left<f_n,X_n\right>_{n<\omega}\subseteq K$ be such that $f_n:\mu\to\mathrm{OR}$ and $X_n\subseteq\mu$ and $X_n\in U$ and $f_{n+1}(\alpha)<f_n(\alpha)$ for all $\alpha\in X_n$, for all $n<\omega$. Let $\vec{X}=\left<X_n\right>_{n<\omega}$. Because $K|\mu^{+K}$ is definable over $\mathcal{H}_\kappa$ and satisfies "$V=\mathrm{HOD}$", each $X_n$ is specified by some ordinal $\alpha<\mu^+$, so there is a relevant $\vec{\alpha}$ such that $\left<X_n\right>_{n<\omega}$ is defined from $\vec{\alpha}$. Let $X=\bigcap_{n<\omega}X_n$. So $X\in\bar{\mathcal{H}}$, and (as $M$ has wellfounded $\omega$) $X\in U$, and in particular $X\neq\emptyset$. But then letting $\beta\in X$, we have $f_{n+1}(\beta)<f_n(\beta)$ for all $n<\omega$, a contradiction.
So $K'=\mathrm{Ult}(K,U)$ is wellfounded, but then the ultrapower map $k:K\to K'$ is non-trivial, a contradiction.
Subcase 1.2: $\mu=\kappa$.
So all $\omega$-sequences $\subseteq\kappa$ are relevant, and it easily follows that $M$ is closed under $\omega$-sequences, and hence wellfounded. We claim that $K\models$"$\kappa$ is weakly compact". For since $\kappa$ is weakly inaccessible, it is inaccessible in $K$. Suppose $\kappa$ is not weakly compact in $K$. We have $K_\kappa=V_\kappa^K\subseteq\bar{\mathcal{H}}$ and $V_{\kappa+1}^K\subseteq\mathcal{H}$, and so $\bar{\mathcal{H}}\models$"$\kappa$ is not weakly compact in $K$". Let $A\in V_{\kappa+1}^K$ with $A\in\bar{\mathcal{H}}$ and $\varphi$ be a $\Pi_1$ formula such that $(V_{\kappa+1}^K,A)\models\varphi$ but there is no $\kappa'<\kappa$ such that $(V_{\kappa'+1}^K,A\cap V_{\kappa'}^K)\models\varphi$. So the same holds for $j(\kappa)$ in $M$. Since $M$ is wellfounded, we have $\kappa\in M$, and can apply the statement at $\kappa'=\kappa$ in $M$, which implies $(V_{\kappa+1}^{K^M},j(A)\cap V_{\kappa}^{K^M})\models\neg\varphi$, and so in fact $(V_{\kappa+1}^{K^M},A)\models\neg\varphi$. But $M$ is closed under $\omega$-sequences, which implies that $K^M$ is iterable, and so $K^M|\kappa^{+K^M}$ is a segment of $K$ so $V_{\kappa+1}^{K^M}\subseteq K^M$, and since $\neg\varphi$ is $\Sigma_1$, therefore $(V_{\kappa+1}^K,A)\models\neg\varphi$, a contradiction.
Now suppose $2^\gamma\leq\kappa$ for all $\gamma<\kappa$. We first claim that $\kappa$ is actually inaccessible. For suppose not, and let $\gamma<\kappa$ be least such that $2^\gamma\geq\kappa$; by our assumption then, $2^\gamma=\kappa$. Thus, we chose $\vec{A}$ to enumerate all subsets of $\gamma$, in a one-to-one fashion. But then $j(\vec{A})$ properly extends $\vec{A}$ with new subsets of $\gamma$, a contradiction. Now one can show that $\kappa$ is weakly compact via an argument just like that used for $K$ above, but now in $V$ (and it is easier).
Case 2: Otherwise (there is $\gamma<\kappa$ such that $\gamma^{\omega}>\kappa$).
This case will be dealt with similarly to the previous one, but it is a little subtler. The relevant sequences are the finite tuples $\vec{\alpha}\in\eta^{<\omega}$, where $\eta=\kappa$ in case (ii.1).
Subcase 2.1: $\mu<\kappa$.
Define $\delta,U$ as in Subcase 1.1. We claim again that $\mathrm{Ult}(K,U)$ is wellfounded, which is again enough. Suppose not and let $\left<f_n,X_n\right>_{n<\omega}\subseteq K$ be as before. We have $X_n\subseteq\mu$. Let $\alpha_n$ be the rank of $X_n$ in the $K$-constructibility order. Then $\alpha_n<\mu^{+K}\leq\mu^+\leq\kappa$. By covering for $K$, there is a set $\mathcal{X}\in K$, of cardinality $\leq\aleph_1^V$, such that $X_n\in\mathcal{X}$ for each $n$, and $Y\subseteq\mu$ for each $Y\in\mathcal{X}$. Since $\aleph_2^V<\mu$, the usual argument shows that $Z=\{Y\in\mathcal{X}\bigm|Y\in U\}$ is in $K$. (That is, consider $\{Y\in j(\mathcal{X})\bigm|\delta\in Y\}\in M$, and use the agreement between $M$ and $\bar{\mathcal{H}}$ below $\mu$.) But then also since $\aleph_2^V<\mu$, it follows that $\bigcap Z\neq\emptyset$. Therefore $\bigcap_{n<\omega}X_n\neq\emptyset$, so now we can argue for contradiction like in Subcase 2.1.
Subcase 2.2: $\mu=\kappa$.
In this case we show that $\kappa$ is weakly compact in $K$.
Suppose (ii.2) holds, so $\kappa^{+K}=\kappa^+$ and $\bar{\mathcal{H}}=\mathrm{Hull}^{\mathcal{H}}(\eta)$, and $\mathrm{cof}(\eta)=\kappa$. Suppose $\kappa$ is not weakly compact in $K$. Then $K|\eta=K^{\bar{\mathcal{H}}}\models$"$\kappa$ is not weakly compact". So fix a counterexample $A,\varphi\in K|\eta$.
Define $U,\delta$ as before. We claim that $\mathrm{Ult}(K|\eta,U)$ is wellfounded. So suppose otherwise and let $\left<f_n,X_n\right>_{n<\omega}$ be a counterexample $\subseteq K|\eta$. Let $\alpha_n$ be the rank of $X_n$ in the order of constructibility of $K$. Let $\mathcal{X}=\{\alpha_n\bigm|n<\omega\}$. So by covering, there is $\mathcal{Y}\subset\mathrm{OR}$ such that $\mathcal{Y}\in K$ and $\mathcal{X}\subseteq\mathcal{Y}$ and $\mathcal{Y}$ has cardinality $\leq\aleph_1^V$. It suffices to see there is such a $\mathcal{Y}\in K|\eta$, as then we can argue as before. Since $\kappa$ is the largest cardinal of $K|\eta$, there are cofinally many $\beta<\eta$ such that $K|\beta$ projects to $\kappa$, which means here that there is a surjection $f:\kappa\to K|\beta$ which is definable without parameters over $K|\beta$. Fix such a $\beta,f$ with $X_n\in K|\beta$ for all $n$; this exists because $\mathrm{cof}(\eta)=\kappa$ and $\kappa$ is weakly inaccessible. So $f\in K|(\beta+1)\subseteq K|\eta$. Let $\mathcal{X}'$ be the set of all $\alpha<\kappa$ such that $f(\alpha)\in\mathcal{X}'$ and for no $\alpha'<\alpha$ is $f(\alpha')=f(\alpha)$. So $\mathcal{X}'\subset\kappa$ and $\mathcal{X}'$ is countable. So by covering, there is $\mathcal{Y}'\in K$ with $|\mathcal{Y}'|\leq\aleph_1^V$ and $\mathcal{X}'\subseteq\mathcal{Y}'$. Since $\kappa$ is weakly inaccessible, $\mathcal{Y}'\in K|\kappa$. But then $\mathcal{Y}=f``\mathcal{Y}'$ covers $\mathcal{X}$ and $\mathcal{Y}\in K|(\beta+1)\subseteq K|\eta$, as desired.
Also, $N=\mathrm{Ult}(K|\eta,U)$ is iterable. For it suffices to see that all countable elementary substructures of $N$ are iterable. For this, given $\left<f_n\right>_{n<\omega}\subseteq K$, it suffices to see that $\bar{N}=\mathrm{Hull}^N(\{[f_n]\bigm|n<\omega\})$ is iterable (where $f_n$ represents $[f_n]\in N$). But we can find sets $X_n\in K|\eta$ such that the $\Sigma_n$-elementary theory of $(f_0(\alpha),\ldots,f_n(\alpha))$ in $K|\eta$ is independent of $\alpha\in X_n$, and then arguing as before, $\bigcap_{n<\omega}X_n\neq\emptyset$, so letting $\alpha\in\bigcap_{n<\omega}X_n$, $\bar{N}$ is isomorphic to $\mathrm{Hull}^{K|\eta}(\{f_n(\alpha)\bigm|n<\omega\})$, and is therefore iterable.
By the iterability, $N|\kappa^{+N}=K|\kappa^{+N}$, and so the failure of weak compactness now leads to a contradiction like before (applying the ultrapower map $i^{K_\eta}_U:K|\eta\to N$).
Finally if (ii.1) holds, it is almost the same as for (ii.2), but easier: note that $\kappa^{+K}<\eta=\mathrm{OR}\cap\bar{\mathcal{H}}$ in this case, so taking any (small) covering set $\mathcal{Y}\in K$ for a countable set $\mathcal{X}$, we get $\mathcal{Y}\in\bar{\mathcal{H}}$, which is again enough.
Regarding specifically $(\kappa,\kappa^{++})$-compactness of $\mathcal{R}$-satisfiability:
Observation 1: Assuming ZFC + $\kappa$ is uncountable + $\mathcal{R}$-satisfiability is $(\kappa,\kappa^{++})$-compact, we get an inner model with a measurable, by arguing much as above, but allowing relevant sequences to take values anywhere ${<\kappa^+}$; this leads to a model $\bar{\mathcal{H}}\preccurlyeq\mathcal{H}$ with $\kappa^+\subseteq\bar{\mathcal{H}}$ and an embedding $j:\bar{\mathcal{H}}\to\widetilde{\mathcal{H}}^M$ at the end, and $j(\kappa)>\kappa$. This then gives an inner model with a measurable like when $\mu<\kappa$ in the preceding proof.
Observation 2: In the converse direction: Assume ZFC + GCH + $\kappa$ is $\kappa^+$-supercompact. Let $G$ be generic over $V$ for adding a $\kappa^{++}$-seqence of Cohen reals with the finite support product. Then $V[G]\models$"$2^{\aleph_0}=\kappa^{++}$ and $\mathcal{R}$-satisfiability is $(\kappa,\kappa^{++})$-compact".
This can be shown using a small variant of the argument in Update 2 of my answer to https://mathoverflow.net/questions/394526/is-this-compactness-property-for-satisfiability-on-mathbbr-consistent, which does the analogous thing for $(\kappa,\kappa^+)$-compactness from weak compactness.