What feature(s) must a (non 1st-order) language with proper-class-many formulas have in order to guarantee that:
There is a proper class P of formulas such that both
(a) every set-sized sub-collection of P is satisfiable, and
(b) no proper-class-sized sub-collection of P is satisifable?
Let me give a few examples.
Example 1. Let us work in Gödel-Bernays set theory, and assume that $T\subset {}^{<\text{Ord}}2$ is a proper class tree of height Ord, but there is no cofinal branch.
(This theory is consistent relative to an inaccessible cardinal, because if $\kappa$ is inaccessible and not weakly compact, then there is a $\kappa$-Aronszajn tree $T\subset {}^{<\kappa}2$, and then $V_\kappa$ with all subsets is a model of GBC where $T$ has the desired property.)
In the logic $L_{\infty,\omega}$, which allows arbitrary sized conjunctions and disjunctions, with a constant for every element of $T$ and a unary predicate symbol $B$, consider the theory $P$ consisting of the assertions $\varphi_\alpha$ asserting first, that there is precisely one object $u$ on level $\alpha$ of the tree that satisfies $B$, and secondly, that in this case, every $v<_T u$ also has $B(v)$. These assertions can be made in the logic $L_{\infty,\omega}$ using constants for the elements of $T$. Thus, altogether, $P$ is the theory asserting that $B$ is a cofinal branch through the tree.
Every set-sized subtheory of $P$ mentions only a bounded number of levels, and so we can find a model by picking any node above that bound and using the predecessors of that node as the instantiation of $B$.
But under our assumptions that the tree $T$ is Ord-Aronszajn, there can be no model of all of $P$ or even of a proper class sized subtheory of $P$, because any such subtheory will involve the assertions concerning unboundedly many levels of $T$, and so the model of that subtheory will pick out a cofinal branch in $T$; but there is no such branch.
Meanwhile, there is a strong connection between your property and (non)weak compactness, because an inaccessible cardinal $\kappa$ is weakly compact just in case we have the $\kappa$-compactness property for $L_{\kappa,\kappa}$ theories of size $\kappa$. (And there are diverse variations on this.)
Example 2. Here is a different kind of related example using only first-order logic.
Theorem. There is a proper class first-order theory $P$, such that every set-sized subtheory of $P$ has a model, but no class is a model of the whole of $P$.
Proof. We interpret this as a theorem scheme in ZFC, where by "class" we mean a definable class (allowing parameters). Thus, I shall provide a definition of a theory $P$, and then prove first, that every set-sized subtheory of $P$ is satisfiable, and second, that no definable class is a model of $P$.
Let $P$ be the theory in the language of set theory $\in$ augmented with a predicate $\newcommand\Tr{\text{Tr}}\Tr$, meant to serve as a truth-predicate, plus a constant for every object in the universe. The theory $P$ asserts that $\Tr$ obeys all instances of the recursive Tarskian truth definition:
For any set many such assertions, we can find a model, since we can find some $V_\theta$ large enough to contain all the parameters mentioned in the subtheory, and then use truth in $\langle V_\theta,\in\rangle$, which will satisfy all the assertions made in the subtheory.
But no definable class can satisfy $P$, because this is exactly the content of Tarski's theorem on the non-definability of truth. QED
Meanwhile, this theory $P$ of the theorem does has proper-class sized subtheories that are satisfiable, since we could, for example, restrict to the quantifier-free assertions; since there are so many constants, we can produce proper class trivially satisfiable subtheories.
