I got a new AT power supply for the old box. Unfortunately, it seems like my assumption that the power supply is the problem was wrong. Both power supplies failed to start. I'm now thinking that the motherboard or cpu is fried. When I hooked up a runs-no-matter-what 486 power supply to the motherboard, the fan whirred, but there was no beep.
I've got less than a week to make this right. I'll try plugging the CPU into the other socket 7 system I have, but it doesn't support MMX, so I don't really know if that's a valid test.