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First of all, the result you're trying to prove isn't true as stated. It is only true if you restrict $\mathcal{C}$ to be the category of $k$-linear colimit-preserving functors. Second, you don't need a natural map $M\otimes_k G(R)\to M$; you need a natural map $M\otimes_k G(R)\to G(M)$. To construct such a map, first consider the case $M=R$. In that ...
We want to show that if $P$ is the projective cover of $M$, then no proper submodule $P'$ of $P$ surjects onto $M$. The key is to know that $Ker(\theta)\ll P$ means $Ker(\theta)$ is a superfluous submodule of $P$, i.e. for all submodules $L\subseteq P$, $P=Ker(\theta)+L$ implies $L=P$. Suppose $P'\subsetneq P$ is a submodule and $P'\twoheadrightarrow M$. We ...