Welp, fortunately a pool repair service had someone available that was 10 min away from me.
They came out, were also unable to open the drain valve. The valve had cracked in the center, and that's where the leak was coming from (I bet someone over tightened it long ago). This is also why the leak got faster after we tried getting the drain plug out, it widened the crack.
They opened up the filter and drilled it out from the inside (I cleaned off my absolutely gross filter cartridges while they were working on that part). I was amazed at how much green those filters were holding.
Pump is back running, with much cleaner filters, and no leaks (actually there's one on the union on the pump outlet, but it only leaks for ~15s when the pump starts up, then goes away).