Let $A$ and $B$ be $k$-algebras with $k$ a field. Suppose that $\psi \colon A \xrightarrow{\cong} B$, and recall that $\mathrm{HH}^*(A) = \mathrm{Ext}_{A^e}^*(A,A)$ with the Gerstenhaber cup product which agrees with the composition product is a graded commutative ring. Also, for each pair of $B$-modules $M$ and $N$, we have an isomorphism $$\varphi_{M,N} \colon \mathrm{Ext}_B^*(M,N) \to \mathrm{Ext}_A^*(M,N) \,.$$ Note that $\mathrm{Ext}_B^*(M, N)$ is a right $\mathrm{HH}^*(B)$-module and that $\mathrm{Ext}_A^*(M, N)$ is a right $\mathrm{HH}^*(A)$-module.
My question is whether there exists $\Psi \colon \mathrm{HH}^*(B) \xrightarrow{\cong} \mathrm{HH}^*(A)$ such that $$\varphi_{M,N}(\beta) \cdot \Psi(\alpha) = \varphi_{M,N}(\beta \cdot \alpha) \quad \text{for each $\alpha \in \mathrm{HH}^*(B)$, $\beta \in \mathrm{Ext}_B^*(M, N)$} \,.$$
If so, could someone suggest a reference or how to get started on showing this?
(Addition: If it’s true in a specific case, like for group algebras or commutative algebras, I’d be interested in that, as well.)