SGA 4 VI Discusses finiteness conditions one can impose on topoi to make limits behave correctly. I am not that familiar with SGA but it is my impression that this expose only discusses abelian sheaves.

Ideally I would like to find written in the literature an analogue of the statement 8.7.7 but in a "non-abelian context" that proves the same thing for $H^1$. Something like the formula: $H^1(\lim F_i, G_i)=\lim H^1(F_i, G_i)$ as pointed sets.

I suppose one could do a careful read of this expose and verify that this holds true, this statement is probably both true and known. I am just wondering if it is written somewhere I can reference.