Ah, turns out the other body is an AE-1 Program. It's missing an exterior piece of leatherette, on the front of the camera, but that's fixable. I moved the new battery to this one and it also seems to meter and fire well. Neither one squeals either.
They were used by the lady I bought 'em from, to create photo mousepads to sell, and they're pretty clean, so I figure they were well maintained. She'd recently gone digital, and seemed happy that someone who was actually interested in film photography showed up to pick 'em up.