Here's the model structure that I think works. Let $\mathcal{C}$ be a cofibrantly generated model category, with generating cofibrations $I$ and generating trivial cofibrations $J$. Let $K$ be a small* category with models $M$ (so $M$ is a set of objects in $K$). We want to construct a cofibrantly generated model structure on $\mathcal{C}^K$, the category of functors $K \to \mathcal{C}$, such that a weak equivalence (resp. fibration) of functors $F \to G$ is one such that $Fm \to Gm$ is one such for $m \in M$.
Namely, let's define fibrations and weak equivalences in that way. Define cofibrations (or trivial cofibrations) by use of the appropriate lifting property. The claim is that if $m \in M$, and $x \to y$ is in $I$ (resp. $J$), then $\hom(m, \cdot) \times x \to \hom(m, \cdot) \times y$ is a cofibration (resp. trivial cofibration) in $\mathcal{C}^K$. This follows because lifting with respect to a diagram $F \to G$ is the same as finding a lifting in the diagram in $\mathcal{C}$ with $x \to y$ versus $Fm \to Gm$ (by Yoneda's lemma). Moreover, we see that these generate the cofibrations (resp. trivial cofibrations) in $\mathcal{C}^K$ in the sense that anything having the right lifting property with respect to these morphisms is necessarily one such that $Fm \to Gm$ is a trivial fibration (resp. fibration) in $\mathcal{C}$ for each $m$, so is a trivial fibration (resp. fibration) in the functor category.
So the lifting axiom is clear. The retract axiom is easy to verify, and the factorization axiom follows from the small object argument applied to the maps $\hom(m, \cdot) \times x \to \hom(m, \cdot) \times y$. This shows that one actually has a model structure.
In short, one can just imitate the projective model structure by loosening the notions of fibration and weak equivalence.When one takes $\mathcal{C}$ to be the category of chain complexes with the usual model structure, then the acyclic model theorem is just the lifting axiom applied to the model structure on $\mathcal{C}^K$, as described in the question.
*There are set-theoretic issues here when $K$ is something like spaces or simplicial sets, of course. I'm not really well-versed enough in the details to know what to do that works reasonably other than appeal to universes (although this is not necessary).