Fix a finitely generated left $R$-module $_RP$ that is in $\mathcal{C}$ and projective as an object of $\mathcal{C}$, and a finite dimensional left $S$-module $_SC$.

As noted in the question, for any $R$-$S$-bimodule $_RM_S$ there is an obvious map
$$\Phi_M:\operatorname{Hom}_R(P,M)\otimes_SC\to\operatorname{Hom}_R(P,M\otimes_SC),$$
and it is natural in $M$.

If $M$ is of the form $X\otimes_\mathbb{C}S$ for $_RX$ a left $R$-module, then $\Phi_M$ is an isomorphism, using the fact that $_RP$ is finitely generated as a left $R$-module, so that the natural map
$$\operatorname{Hom}_R(P,X)\otimes_\mathbb{C}V\to
\operatorname{Hom}_R(P,X\otimes_\mathbb{C}V)$$ is an isomorphism for any vector space $V$.

The standard projective bimodule presentation
$$S\otimes_\mathbb{C}S\otimes_\mathbb{C}S\to S\otimes_\mathbb{C}S\to S\to0$$
of $S$ induces an exact sequence
$$\tag{*}
B\otimes_\mathbb{C}S\otimes_\mathbb{C}S\to
B\otimes_\mathbb{C}S\to B\to0,$$
which is split as an exact sequence of left $R$-modules, so that applying the functor $M\mapsto\operatorname{Hom}_R(P,M)\otimes_S C$ to it preserves exactness.

Also, applying $-\otimes_SC$ to the exact sequence $(*)$ gives an exact sequence
$$\tag{**}
B\otimes_\mathbb{C}S\otimes_\mathbb{C}C\to
B\otimes_\mathbb{C}C\to B\otimes_SC\to0.
$$

As a sequence of left $R$-modules, the first term is a (possibly infinite) direct sum $\bigoplus_{i\in I}B$ of copies of $B$ and the second is (since $C$ is finite dimensional) a finite direct sum of copies of $B$.

This is the filtered colimit, over finite subsets $J\subseteq I$, of exact sequences of the form
$$\tag{***}
\bigoplus_{j\in J}B\to B\otimes_\mathbb{C}C\to U_J\to 0,$$
where the first two terms are in $\mathbb{C}$ since $_RB$ is in $\mathcal{C}$, and since $\mathcal{C}$ is an abelian subcategory, the third term is also in $\mathcal{C}$. Therefore $\operatorname{Hom}_R(P,-)$ preserves exactness of the sequences $(***)$ since $P$ is projective in $\mathcal{C}$. Since $P$ is finitely generated as an $R$-module,
$\operatorname{Hom}_R(P,-)$ also preserves filtered colimits, and so applying $\operatorname{Hom}_R(P,-)$ to $(**)$ also gives an exact sequence.

So applying
$$\Phi:\operatorname{Hom}_R(P,-)\otimes_SC\to\operatorname{Hom}_R(P,-\otimes_SC)$$
to the sequence $(*)$ gives a map of exact sequences which is an isomorphism on the first two terms and therefore also on the third term.

So $\Phi_B$ is an isomorphism.

This doesn't seem to require $B\otimes_SC$ to be finite dimensional.

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