An isomorphism concerned about any finitely generated projective module

Hint: This is where you need to use finite generation and projectiveness of $P$. Since $P$ is finitely generated, there is a surjective $R$-map $p : R^n \to P$ for some $n > 0$. Projectiveness of $P$ yields a section $s : P \to R^n$ of $p$ (so that $p\circ s = \operatorname{id}_P$). For $1\le i\le n$, let $s_i$ be the composition $P \xrightarrow{s} R^n \xrightarrow{\pi_i} R$, where $\pi_i$ is the projection onto the $i$th coordinate. Now we define an $R$-map $$\operatorname{Hom}(P,M) \to \operatorname{Hom}(P,R) \otimes M$$ by sending a $g\in \operatorname{Hom}(P,M)$ to $\sum s_i \otimes g(p(e_i))$, where $e_1,\ldots, e_n$ is the standard basis of $R^n$.