OK, so as a last ditch effort, I bought a conductive paint pen to go over some of the traces that were close enough to the battery to be possibly affected by the corrosion. While they did not appear to me to be damaged, apparently they were. Going over them with the conductive paint fixed the problem.

It ain't pretty but the board is working again. Time will tell if the board is as stable as it was before and whether the fix will last.

Thanks to everyone for their input!

Way to go. It's a satisfying feeling when you finally get something working again. I hope the motherboard works out well for you