Jigsaw with aggressive blade. Mark the piece to be removed; first cut
is continuous from entry-to-exit with turns at the corners. Clean up
the corners after the bulk of the removal is complete.
If it does not have to look nice, make the corner turns wider such that
everything goes on the first cut and the bottom of the cut is wider
than it needs to be.
If it needs to be "better" than a jigsaw, how does it need to be
"better"
(other than not requiring a jigsaw purchase)?