If the algebraic group is simple, say over an algebraically closed $k$, then one has the following lemma.
Lemma. If $V$, $W$ are finite dimensional, there is an $m$ depending on $V$, $W$, so that if $St_n$ is the $n$-th Steinberg module with $n\geq m$ the natural map $Ext^i(V,W)\to Ext^i(V,W\otimes St\otimes St)$ vanishes.
See for instance my 1977 joint paper with Cline, Parshall and Scott. So this gives a way to kill extension classes while staying within the category of finite dimensional representations. Now it seems one may argue as in sasha's answer, dualized.
For reductive groups over an algebraically closed $k$ the Lemma still holds, interpreted appropriately. But for nonreductive groups one would need something different.