When does the natural simplicial enrichment of the category of cdgas compute the derived mapping space?

Question

Let $CDGA$ be the category of commutative differential graded algebras over a field $k$ of characteristic zero. Denote by $\Omega\left(\Delta^n\right)$ the cdga of algebraic differential forms on the $n$-simplex. There is a natural simplicial enrichment of CDGA given by $Map_n\left(A,B\right)=Hom(A,B \otimes \Omega\left(\Delta^n\right))$ which computes the correct derived mapping space (for the projective model structure) whenever $A$ is quasi-free (i.e. free with a possibly non-trivial differential), or more generally, when $A$ is projectively cofibrant. The main ingredient in showing this is showing that for any CDGA $A,$ if $A \to C$ is a quasi-free $A$-algebra, then the induced map $$Map(C,B) \to Map(A,B)$$ is a trivial Kan fibration. (This is true more generally for any trivial cofibration with respect to the projective model structure, but these are all retracts of quasi-free maps.) My question is, are there more general maps other than trivial cofibrations which get sent to trivial Kan fibrations by $Map(blank,B)$? For instance, is this more generally true for any map $A \to C$ which as a map of graded algebras is of the form $A \to A \otimes K,$ with $K$ acyclic (but not necessarily free)?