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 an $R$-$S$-bimodule $_RB_S$ there is an obvious map
$$\Phi_B:\operatorname{Hom}_R(P,B)\otimes_SC\to\operatorname{Hom}_R(P,B\otimes_SC),$$ 
and it is natural in $B$.

Let $\mathcal{B}$ be the category of $R$-$S$-bimodules $_RB_S$ such that $_RB$ is in $\mathcal{C}$. Since $\mathcal{C}$ is an abelian subcategory of the category of left $R$-modules, $\mathcal{B}$ is an abelian subcategory of the category of $R$-$S$-bimodules. 

Every object $B$ of $\mathcal{B}$ is a quotient of an object of the form $X\otimes_\mathbb{C}S$, where $_RX$ is an object of $\mathcal{C}$, since there is an epimorphism $b\otimes s\mapsto bs$
$$_RB\otimes_\mathbb{C}S_S\to B$$
of $R$-$S$-bimodules. 

So for any $B$, there is an exact sequence
$$X'\otimes_\mathbb{C}S\to X\otimes_\mathbb{C}S\to B\to0,$$
where $X$ and $X'$ are in $\mathcal{C}$,
inducing an exact sequence
$$X'\otimes_\mathbb{C}S\otimes_SC\to X\otimes_\mathbb{C}S\otimes_SC\to B\otimes_SC\to0.$$
Since the first two terms are in $\mathcal{C}$, which is an abelian subcategory of the category of left $R$-modules, so is $B\otimes_SC$.

Let $\mathcal{B}'$ be the full subcategory of $\mathcal{B}$ consisting of the objects $B$ for which $\Phi_B$ is an isomorphism.

Since the functors $B\mapsto\operatorname{Hom}_R(P,B)\otimes_SC$ and $B\mapsto\operatorname{Hom}_R(P,B\otimes_SC)$ are both right exact functors from $\mathcal{B}$ to vector spaces (since $P$ is projective in $\mathcal{C}$), the category $\mathcal{B}'$ is closed under cokernels.

Also, $\mathcal{B}'$ contains all objects of the form $X\otimes_\mathbb{C}S$ for $X$ in $\mathcal{C}$, using the fact that $_RP$ is finitely generated as a left $R$-module, so that the natural map $$\operatorname{Hom}_R(P,X)\otimes_\mathbb{C}S\to\operatorname{Hom}_R(P,X\otimes_\mathbb{C}S)$$
is an isomorphism.

So, since we know that every object of $\mathcal{B}$ is the cokernel of a map between such objects, $\mathcal{B}'=\mathcal{B}$.