A condition for the validity of an identity in the derived category of bounded complexes of modules.

Question

Let $\mathcal{R}$ be a sheaf of (not necessarily commutative) rings on a topological space $X$ and let $M$ and $N$ be $\mathcal{R}-$modules such that $M$ is quasi-isomorphic to a finite complex of locally free $\mathcal{R}$-modules of finite rank. Can we infer from these assumptions that there is an isomorphism $$R\mathcal{Hom}_{\mathcal{R}}(M, \mathcal{R})\otimes_{\mathcal{R}}^LN\cong R\mathcal{Hom}_{\mathcal{R}}(M, N)$$ in the derived category $D^b(\mathcal{R})$ of bounded $\mathcal{R}$-modules? In particular, do we need to assume in addition that $M$ is a coherent $\mathcal{R}$-module or the existence of a finite resolution of locally free $\mathcal{R}$-modules of finite rank suffices?