Is this NP-completeness proof correct?
In your specific case, what other constraints do you have on $M$ and $N$? For example, if it is possible that $M=N=1$, then each of your clauses will have 2 literals, and your SAT problem actually seems to become the 2-SAT problem which is in P.

So if $P_1$ depends on $M$ and $N$, maybe you should consider an NP-complete SAT variant that somehow takes them into account. In general, if you want to add some additional constraints on SAT (or any other problem you reduce from), make sure the problem still remains NP-complete.