Here is the construction in the case of limits.
A diagram $(X, a) : \mathsf{J} \to \int F$ consists of a diagram $X = \Pi \cdot (X, a) : \mathsf{J} \to \mathsf{C}$ along with a cone $a : * \to F\cdot X$ in $\mathsf{Set}$. Suppose that $X : \mathsf{J} \to \int F \to \mathsf{C}$ has a limit cone $\eta : \lim X \to X$ and that $F(\eta) : F(\lim X) \to F \cdot X$ is also a limit cone.
We'll show that if $\eta$ has a lift then the lift is unique. Suppose that $\eta$ lifts to a cone $\tilde{\eta} : z \to (X, a)$. Since $\Pi$ acts as the identity on arrows we must have $\tilde{\eta}_{j} = \eta_{j}$ for all $j$ in $\mathsf{J}$. So we must have $\Pi(z) = \lim X$, meaning that $z$ is of the form $(\lim X, x)$, where $x$ is an element of $F(\lim X)$. Since $\tilde{\eta} : (\lim X, x) \to (X, a)$ is a cone in $\int F$ we must have $F(\eta_{j})(x) = a_{j}$ for all $j \in \mathsf{J}$. And since $F(\eta)$ is a limit cone there is a unique element $x$ of $F(\lim X)$ satisfying these equations, so there is at most one lift of $\eta : \lim X \to X$ to a cone in $\int F$.
So now to show that $\Pi$ strictly creates limits we just have to check that this $\tilde{\eta} : (\lim X, x) \to (X, a)$ is a cone in $\int F$ and that it is limiting. I'll leave it to you to do the checking.
There are several "size" issues involved here.
One is literally about size: some of the collections involved have too many elements to be small.
But another is more about complexity: some collections have a small number of elements but those elements themselves are not small – such collections do not even exist in the ontology of NBG or MK, in the same way that classes do not really exist in the ontology of ZFC.
Complexity issues can be resolved by using a set theory with a richer ontology, such as ZFC + universes, but size issues in the narrow sense cannot be worked around so easily.
To be concrete, I will work in ZFC + universes, but what I say applies more generally.
Let $\mathcal{A}$ be a category, let $\textbf{Set}$ be the category of small sets, and let $\textbf{Psh} (\mathcal{A})$ be the category of all functors $\mathcal{A}^\textrm{op} \to \textbf{Set}$.
If $\mathcal{A}$ is small, then functors $\mathcal{A}^\textrm{op} \to \textbf{Set}$ can be implemented as small sets, and likewise natural transformations between them, so $\textbf{Psh} (\mathcal{A})$ is a locally small category of similar complexity as $\textbf{Set}$.
If $\mathcal{A}$ is essentially small, then $\textbf{Psh} (\mathcal{A})$ is equivalent to a locally small category, but possibly not a locally small category in the strictest sense, for complexity reasons.
If $\mathcal{A}$ is not even essentially small, then $\textbf{Psh} (\mathcal{A})$ may fail to be locally small even up to equivalence.
Anyway, for any category $\mathcal{C}$ whatsoever:
Proposition.
Let $X : \mathcal{I} \to [\mathcal{A}^\textrm{op}, \mathcal{C}]$ be a diagram.
If, for each object $A$ in $\mathcal{A}$, the diagram $\textrm{ev}_A \circ X : \mathcal{I} \to \mathcal{C}$ has a limit (resp. colimit), then $X : \mathcal{I} \to [\mathcal{A}^\textrm{op}, \mathcal{C}]$ has a limit (resp. colimit) and, for all objects $A$ in $\mathcal{A}$, $\textrm{ev}_A : [\mathcal{A}^\textrm{op}, \mathcal{C}] \to \mathcal{C}$ preserves that limit (resp. colimit).
The proof is straightforward and constructive (provided the hypotheses are understood constructively, i.e. we are given a (co)limit (co)cone for each $\textrm{ev}_A \circ X$ and not mere existence).
Thus it is robust and there are basically no size issues involved.
Where size issues (in the narrow sense) start to get involved is when we are less careful about balancing the hypotheses and the conclusions.
For example:
Proposition.
If $\mathcal{C}$ has copowers (resp. powers) indexed by small sets and $\mathcal{A}$ is locally small, then for all objects $A$ in $\mathcal{A}$, $\textrm{ev}_A : [\mathcal{A}^\textrm{op}, \mathcal{C}] \to \mathcal{C}$ preserves limits (resp. colimits) for all (not necessarily small) diagrams.
Notice that the proposition concludes something about arbitrary limits in $[\mathcal{A}^\textrm{op}, \mathcal{C}]$ only assuming hypotheses on small copowers in $\mathcal{C}$.
How can this be?
The answer is that the proof uses a trick: it constructs a left adjoint for $\textrm{ev}_A$, and we know that right adjoints have extremely strong limit preservation properties.
Moreover, if you examine the proof you will realise that the size hypotheses are not optimal: it suffices that $\mathcal{C}$ have copowers indexed by the hom-sets of $\mathcal{A}$.
So, for example, if $\mathcal{A}$ is a locally finite category and $\mathcal{C}$ has finite copowers then $\textrm{ev}_A : [\mathcal{A}^\textrm{op}, \mathcal{C}] \to \mathcal{C}$ preserves arbitrary limits.
Or if $\mathcal{A}$ is a preorder category and $\mathcal{C}$ has an initial object then $\textrm{ev}_A : [\mathcal{A}^\textrm{op}, \mathcal{C}] \to \mathcal{C}$ preserves arbitrary limits.
Best Answer
Yes. A natural transformation between functors $F$ and $G$ is just an element of $\prod\limits_{A \in J} Hom(FA, GA)$ satisfying the naturality condition. So the class of natural transformations from $F$ to $G$ is a subclass of the set $\prod\limits_{A \in J} Hom(FA, GA)$, hence is a set.
Let’s start by assuming $J$ is small and $C$ is locally small. Note that representations of a functor $H : C^{op} \to Set$ are in natural bijection with objects $A$, together with some $a \in HA$, such that for all $b \in HB$ there is a unique $f : Hom(B, A)$ such that $b = H(f)(a)$. This follows from the Yoneda lemma.
In this case, a representation of the functor $Hom(\Delta_J(-), F)$ corresponds to an object $A \in C$, together with some natural transformation $\pi : Hom(\Delta_J(A), F)$ such that for all $b \in Hom(\Delta_J(B), F)$, there exists a unique $f : B \to A$ such that $b = \pi \circ \Delta_J(f)$.
It turns out that we can now define the limit to be such an object $A$ together with such a natural transformation $\pi$. This definition works even when $J$ and $C$ are not assumed locally small. Furthermore, it’s fairly straightforward to prove that limits are unique. For if we had two limits $(A, \pi)$ and $(B, p)$, we could construct back-and-forth maps $f : Hom(A, B)$ and $g : Hom(B, A)$ which are inverse isomorphisms and which preserve the $\pi/p$ structure.
Note that phrasing this will involve quantifying over a collection of proper classes. So even stating the definition of limits over large diagrams requires a meta theory like NBG (a class theory which is a “conservative extension” of ZFC - anything provable about sets in NGB is also provable in ZFC) or MK (which is much, much stronger than ZFC). An alternate tactic is to use Grothendieck universes, which allows you to keep using set theory but requires the adoption of some very strong large cardinal axioms.