In Quillen's monograph Homotopical algebra where he introduced the notion of model category, he showed that if $C$ is a bicomplete category with enough regular-projectives in which either (*) every object is a group object (roughly), or (*) there is a set of small projective generators, then the category $sC$ of simplicial objects in $C$ is a model category with the weak equivalences and fibrations detected by mapping out of projective ones. The hypotheses (*) or (*) are used only to construct the trivial cofibration / fibration factorization.

Now if $C$ is monadic over $\mathrm{Set}$, then it is automatically bicomplete with enough regular-projectives (the regular epis are the surjections and free objects are projective), and if the monad is accessible then the category $C$ satisfies (**). But what if the monad is not accessible?

In particular, I am curious about $C=$ the category of suplattices (posets with all suprema, and suprema-preserving maps). These are the algebras for the covariant powerset monad on $\mathrm{Set}$, which is not accessible. And indeed, there are essentially no nontrivial small suplattices (in the technical sense that mapping out of them preserves sufficiently filtered colimits), so (**) doesn't hold.

So my question is: is there a model structure on the category of simplicial suplattices in which the weak equivalences and fibrations are created in simplicial sets? (I believe this is equivalent to Quillen's definition in this case.) In particular, does every simplicial suplattice have a fibrant replacement?