I think I might have fixed this by cleaning the battery compartment and reseating the battery so thanks for making me take a closer look at that.
I got to put it to the test right after as I thought somehow my entire machine had died (it wouldn't POST; a few times it'd just beep at me but then it wouldn't even do that until I took out the RAM and had them swap places - then it worked). So I had to unplug it a fair number of times and when I got it back up and running it didn't tell me it had lost any BIOS settings (useful since, besides the clock, I need it to keep track of the dual floppy drives I use in it properly).