There is some evidence from characters that $H^4(M,\mathbb{Z})$ contains $\mathbb{Z}/12\mathbb{Z}$. In particular, the conjugacy class 24J (made from certain elements of order 24) has a character of level 288, and the corresponding irreducible twisted modules have a character whose expansion is in powers of $q^{1/288}$. Fusion then yields a $1/12$ discrepancy in $L_0$-eigenvalues (meaning you will pick up 12th roots of unity from the associator). This suggests that pulling back along a pointed map $B(\mathbb{Z}/24\mathbb{Z}) \to BM$ corresponding to an element in class 24J yields a cocycle of order 12. I don't know how the cocycles corresponding to different cyclic groups fit together.
I don't know if you've seen Mason's paper, Orbifold conformal field theory and cohomology of the monster, but it is about related stuff. I don't understand how he got his meta-theorem at the end, though.