Large cardinals, small violations of choice

We study the relationship between large cardinals and Blass' Small Violation of Choice ($\mathsf{SVC}$), which states that the Axiom of Choice holds in a set-generic extension of the universe. Our main theorem is that in the presence of a supercompact cardinal, either the Axiom of Choice fails significantly, or else $\mathsf{SVC}$ holds in $L(\mathcal P(\mathrm{Ord}))$. We show that a supercompact is not strong enough to reflect $\mathsf{SVC}$, while an extendible cardinal is. If we assume Usuba's Loś Property holds for the supercompact, then we can outright derive $\mathsf{SVC}$ in the universe.