I cannot comment due to reputation and haven't done the computations, but I see two possible problems:
usually one has a spectral sequence $(R^if_{*})(\mathcal{E}xt^j(M,N)) => \mathcal{E}xt_f^{i+j}(M,N)$ because $f_{*}\mathcal{H}om(M,-)$ is a composition of functors.
$Ext^2(F,F)=H^2(\mathcal{H}om(F,F))$ only holds if $F$ is locally free, due to local to global spectral sequence for $Ext$.
What may be helpful: there is a base change theorem for relative Ext-sheaves, ma be this can help you