As discussed in the comments, the notion of a set which a theory proves exists is inherently problematic, and it's difficult to formulate a version of the requirement "All sets which we can prove exist, can be proved to be in $L$" which doesn't fail for silly reasons.
It seems to me, though, that the following is probably the strongest theory which will match the intuitions behind such a requirement:
First, we take the usual axioms of ZFC and relativize them to $L$. So, for example, Powerset becomes "For every constructible $x$ there is a constructible $y$ such that for every constructible $z$, if every constructible element of $z$ is in $x$ then $z$ is in $y$, and every constructible element of a constructible element of $y$ is in $x$."
Now the resulting theory $ZFC^L$ is fine: any model of it has a definable subset which is a model of ZFC+V=L. However, by relativizing everything we've made stuff a bit weird. For example, the rest of such a model could be truly awful, and the "$L$-part" itself might not sit nicely in the whole (e.g. it might not be transitive). So we probably want to pass to a stronger theory $ZFC^L_+$, consisting of $ZFC^+$ together with unrelativized Extensionality, Union, Pairing, and Foundation, and an axiom asserting that the ordinals of $L$ are exactly the ordinals of the universe.
The result is a theory all of whose models satisfy a very weak set theory, but which have an inner model satisfying ZFC+V=L. And this seems as close to what you want as I can think of.
Such an object is complicated to describe, but not too complicated. In general, the appearance of reals throughout $L$ is technical but not mysterious: we sort of keep using the same basic tricks over and over again. Standard go-to's include countability witnesses and first-order theories of countable levels of $L$ and related structures; common techniques include Lowenheim-Skolem, the condensation lemma (and the Mostowski collapse), and the use of the $L$-ordering to eliminate parameters.
First, there is a general approach that applies more-or-less to every countable ordinal. Whenever $\alpha$ is countable, so is $L_\alpha$, which means there is a (not unique of course) relation $R\subseteq\omega^2$ such that $(\omega; R)\cong (L_\alpha;\in)$ (I'm assuming $\alpha$ is infinite, here). However, it's easy to see that such an $R$ can never, itself, be in $L_\alpha$. That is, for every countable $\alpha$ there are reals which code bijections between $L_{\beta_0}$ and $\omega$, none of which are in $L_\alpha$, and particular this is true for $\alpha=\beta_0$.
We can further identify a specific such real (using $\alpha$ as a parameter): the least real with respect to the parameter-freely-definable well-ordering of $L$ which codes a bijection between $\omega$ and $L_\alpha$. In case $\alpha$ itself is parameter-freely definable - as $\beta_0$ is - this real is also parameter-freely definable. (We can also give a quick complexity analysis: for ordinals such as $\beta_0$ corresponding to the first level of $L$ satisfying a given first-order theory, the resulting definition is $\Delta^1_2$.)
A more specific argument would be to observe that - conflating a transitive set $A$ with the corresponding $\{\in\}$-structure $(A; \in\upharpoonright A)$ - the structure $L_{\beta_0}$ happens to be a pointwise definable structure; that is, each element in it is definable without parameters in it. This means that $Th(L_{\beta_0})$, the set of Godel numbers of all $\{\in\}$-sentences which are true in $L_{\beta_0}$, is not itself an element of $L_{\beta_0}$.
But this relies on particular properties of $\beta_0$; there are many countable ordinals $\gamma$ such that $L_\gamma$ is not pointwise-definable; indeed, most countable ordinals have this property, in the sense that the set of $\gamma$ such that $L_\gamma$ is not pointwise definable is club. Such an $L_\gamma$ can indeed contain its theory as an element, avoiding Tarski by way of that specific element not being parameter-freely definable. For example, $L_{\omega_1}$ contains every real in $L$, including (since $L$ computes first-order theories correctly) the theory of $L_{\omega_1}$ itself. And we can bring this down to the countable realm too, by applying Lowenheim-Skolem, Mostowski collapse, and condensation to get a countable $\gamma$ such that $L_\gamma\equiv L_{\omega_1}$ and $Th(L_{\omega_1})\in L_\gamma$ (hence $Th(L_\gamma)\in L_\gamma$ since $L_\gamma\equiv L_{\omega_1}$).
Incidentally, if you're not already familiar with it you'll probably be interested in the paper "Gaps in the constructible universe" by Marek and Srebrny.
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EDIT: to my chagrin, the notion of "$n$-admissibility" is not what I thought it was! What I wanted was $\Sigma_n$-admissibility. You can find the definition of $n$-admissibles here; they are vastly smaller than their $\Sigma_n$ counterparts, and indeed for each $n$ the least $n$-admissible is less than the least $\Sigma_2$-admissible. Now $n$-admissibility is a rare notion these days and I've seen "$n$-admissible" used for "$\Sigma_n$-admissible before, but given the relevance of older papers to this topic it's probably a good idea for me to not butcher this distinction.
All limit levels of $L$ beyond $\omega$ satisfy the "basic" axioms (extensionality, foundation, pairing, union) plus choice trivially; once you drop powerset, you're really just talking about replacement/separation.
An infinite limit ordinal $\alpha$ is $\Sigma_n$-admissible if $L_\alpha$ satisfies $\Sigma_n$ replacement; $\Sigma_1$-admissibility is just normal admissibility, and one way to define $\omega_1^{CK}$ is as the smallest admissible ordinal $>\omega$.
So what you're looking for is just the smallest $\Sigma_\omega$-admissible ordinal, where an ordinal is $\Sigma_\omega$-admissible if it's $n$-admissible for each $n$.
Now it's worth stressing that the smallest $\Sigma_{n+1}$-admissible is much, much, much larger than the smallest $\Sigma_n$-admissible. The first admissible limit of admissibles is extremely large (much bigger than the first limit of admissibles) and is still drastically smaller than the first $\Sigma_2$-admissible.
Incidentally, note that ZFC proves that $L_{\omega_1}$ (exists and) is a model of all of ZFC except powerset. From this we get $\beta<\lambda$, and indeed $\beta<(\omega_1)^{L_\lambda}$ (inside $L_\lambda$, take the Mostowski collapse of a countable elementary submodel of $(L_{\omega_1})^{L_\lambda}$). Put another way:
(Keep in mind that $L_\lambda\models$ "There is no largest cardinal," so this is really quite a size difference!)