I am in the following situation. I have two (rather explicit and specific) dg commutative algebras $R,S$ over a field of characteristic $0$. In fact, $S$ is an $R$-algebra, in that I have a map $R \to S$. Because I am interested in computing some derived tensor products $S \otimes_R$, I have worked out a "Koszul" resolution $\tilde S$ of $S$ over $R$. So all seems well.
But! Actually, $S$ and $R$ are both Gerstenhaber algebras (in dg vector spaces; the differential and the Gerstenhaber bracket point in opposite directions), and the map $R \to S$ is a homomorphism of Gerstenhaber algebras. My problem is that I have been so far unsuccessful at giving the resolution $\tilde S$ a Gerstenhaber structure such that the resolved map $R \to \tilde S$ is a homomorphism of Gerstenhaber algebras.
This leads me to two questions. The second question depends on the answer to the first.
Question 1: Does there necessarily exist a resolution of $S$ that computes the derived $S\otimes_R$ and that is Gerstenhaber in a compatible way?
Question 2 if the answer to 1 is YES: How do I construct it?
Question 2 if the answer to 1 is NO: Certainly my homotopy equivalence $S \leftrightarrow \tilde S$ allows me to move the Gerstenhaber structure on $S$ to something on $\tilde S$. What structure on $\tilde S$ does it move to?