Nobody has answered the actual question, though of course the preface to the question should say "any finitely generated commutative monoid is finitely presented in the sense that there is a coequalizer diagram $P_1 stackrel{to}{to} P_0 to M$ with $P_1$ and $P_0$ free commutative and finitely generated."
I believe the answer to the question is no, we cannot always find a presentation $P_1 to P_0 to M$ where $P_1$ maps to the kernel of $P_0 to M$.
However, I don't have a proof.
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