Short version
No, there isn't! "$T$ is model-theoretically-conservative over $S$" is a $\Pi_2$ property, and in general (and in this specific case) analysis of such properties and their approximations land us squarely in the realm of large cardinals.
Longer version:
Preliminaries
As stated, this question requires a background metatheory which is rich enough to directly talk about proper class sized "structures." However, $\mathsf{ZFC}$ is not rich enough to do this directly, and once we do move to such a metatheory there's no real difference between the set/class distinction and the size-$<\kappa$/size-$\kappa$ distinction for a "reasonably closed" cardinal $\kappa$. So to keep things simple, I'm going to modify the question to be about how notions of model-theoretic conservativity vary as we allow larger and larger models.
This is certainly a reasonable thing to worry about! For instance, let $T$ be a (complete, no finite models, finite language) countably categorical theory which is not uncountably categorical. Then the theory $T'$ describing a pair of models of $T$ and the theory $T^\star$ which expands $T'$ by adding an isomorphism between the two models of $T$ are related in the following way: every countable model of $T'$ can be expanded to a model of $T^\star$, but not every uncountable model of $T'$ can be so expanded. This motivates the following definition, which is not standard terminology:
Suppose $A\subseteq B$ are deductively closed theories and $\kappa$ is a a cardinal. Say that $B$ is a $\kappa$-MTC extension of $A$ iff every model of $A$ of size $<\kappa$ can be expaned to a model of $B$.
(The "$<\kappa$" where you may expect a "$\le \kappa$" is a common set-theoretic subtlety; think about e.g. $\aleph_\omega$-MTC vs. $\aleph_{\omega+1}$-MTC. This is a very minor point, though, so feel free to ignore it.) The example above shows that we can have an $\aleph_1$-MTC extension which is not an $\aleph_2$-MTC extension. On the other hand, Morley's theorem - that a countable theory categorical in some uncountable cardinality is categorical in every uncountable cardinality - makes it tricky to push this higher, especially if we avoid large languages. So you might expect only "small" cardinals to be relevant to model-theoretic conservation.
As it turns out, though, the situation is much nastier than that. Let's take the theory $\mathsf{ZFC}$ itself, and consider the theory $T$ (in the language gotten by adding to the language of set theory a unary relation symbol $U$) which adds to $\mathsf{ZFC}$ the single axiom $$\exists xU(x)\wedge\forall y\exists z(U(y)\rightarrow U(z)\wedge z\in y).$$ Then $T$ is a model-theoretically-conservative extension of $\mathsf{ZFC}$ iff $\mathsf{ZFC}$ has no well-founded model.
But we can make this even worse! We could replace the sentence above with the disjunction of that sentence and (allowing some abbreviation) $$\exists x(U\subseteq x\wedge \forall y(U\not=y)).$$ Call the result $S$. Then $S$ is a model-theoretically-conservative extension of $\mathsf{ZFC}$ iff there are no well-founded models of $\mathsf{ZFC}$ which are correct about powerset, i.e. worldly cardinals. And we can continue well up the large cardinal hierarchy:
Let $\lambda$ be the least measurable cardinal. Then there is a pair of countable theories $T\subseteq S$ such that $S$ is a $\lambda$-MTC extension of $T$ but $S$ is not a $\lambda^+$-MTC extesnion of $T$.
And this really uses nothing special about measurability, simply the fact that "$\lambda$ is measurable" is "detected" in the cumulative hierarchy not too high above $\lambda$ (= in $V_{\lambda+2}$). At a glance, the upper bound on this phenomenon - the point where MTC-ness up to that point is equivalent to full model-theoretic-conservativity - is a strongly compact cardinal. Which is bonkers gigantic (and also arises if we ask similar questions where the domain of the universe, rather than the language, is extended - see here)! It also helps us understand the analogue of this question for logics other than FOL (where the proof-theoretic notion of conservativity may break down but $\kappa$-MTC-ness is still a priori meaningful): Makowsky proved that Vopenka's principle guarantees (and is equivalent to) having the analogues of strongly compact cardinals for "every" logic (e.g. the second-order analogue of a strongly compact is an extendible cardinal), so VP is in a sense the axiom that says that MTC-ness is "bounded" for every logic.
