Corroded Sink Drain Cover Removal Suggestions
Use a "mini-hack" saw, they work great for actually slicing the drain spud then grabbing it with Channellocks, crushing it, then pulling it out.

Or you could attack from below with a long-shank sacrificial slotted screwdriver and bust the corroded nut off by pounding the screwdriver with a hammer and using it like a chisel.

Brute force carefully applied will do the trick, I've done it many times.