The case of Fano varieties is simpler than the KMM theorem, by the Kodaira vanishing theorem $h^{p,0}=0$ for $p<\dim M$ and then Serre duality gives $h^{0,p}=0$ for $p>0$.

Mea culpa. My comment was wrong in almost any respect. I leave it both in order to leave the subsequent comment meaningful and as a reminder to myself.

I have the impression that Gröbner basis algortihms are very tricky to implement efficiently and if so this may put too much emphasis on gory details.

I think the question may be based on the fact that for non-finitely generated modules projective and locally finite are not the same thing. Take for instance an elliptic curve (minus the origin to make it affine) and take the sum of all the line bundles (one for each point on the original elliptic curve). Then this sum is projective but there is no non-empty open subset of the spectrum over which it is free.

The exercise is no doubt referring to the higher cohomology of the structure sheaf. The holomorphic Lefschetz fixed point formula shows that if its higher cohomology vanishes then any endomorphism has fixed points. This also answers the question as even just the vanishing of $h^{0,p}$ for $p>0$ ensures the existence of a fixed point.

Another proof is in Curtis-Reiner: Representation theory of finite groups and associative algebras, Thm 32.9 in the first edition. To my mind the proof is very cute, it considers the generating series $\sum_k a_k t^k$, where $a_k$ is the number of times the irreducible representation occurs in the $k$'th tensor power of $\rho$. Using the orthogonality formulas for characters they sum this up as a rational function which, because $\rho$ is faithful, has a simple pole at $t=1/n$ and hence many of the $a_k$ are non-zero.

To begin with are you interested in rational or integral cohomology? Secondly, the distance to the $\mathrm{P}^1$ case is perhaps not that great. For a general smooth curve $C$ you have a map from your moduli stack to the $m$'th symmetric power of $C$ associating to a sheaf its cycle. Étale locally this map is independent of the curve and you might try to use the Leray spectral sequence for it.