OK ... just thought I'd update this thread in case someone else ever has this problem.
1) I checked all three batteries (with a multi-meter) that I've tried in this motherboard since they all came from the same package ... all are OK at 3 volts and work fine in other PC's.
2) I gently cleaned the two metal battery holder contacts with a pencil eraser ... and inspected the contacts for any bends. Both contacts seem OK and push up nicely against the negative side of the battery while the battery is in the slot (so seems like good battery contacts).
3) I left the PSU plugged into power when the PC is off to see if the CMOS still loses time (it does). I had read that if you leave the PSU plugged into the house power ... that it sometimes continues to provide power to the motherboard and battery ... allowing the CMOS clock to keep it's data (time, etc).
4) I checked one more time the behavior of the CMOS time when the PC is turned off. It pretty reliably loses a minute if you shut it off and turn it back on in a minute. And same for any other shut off time interval (off for 30 minutes, 4 hours, etc). Also loses date after 24 hours.
So I'm thinking I either need to upgrade the BIOS (I'm still trying to find one that I feel confidant is for this old motherboard) ... or ... I may need to replace the RTC crystal. I've never done that before ... but I see it's near the CMOS chip. I may try to find a crystal on another motherboard if it is of the correct values. I'm leaning towards replacing the RTC crystal at some point.
Anyway ... I forgot how much fun it was to try to bring these old PC's back to life 😀