....snip...
Some PCs have non-rechargeable system button cells that can simply be
replaced.

Sounds like you have a bad NiCAD/NiH battery and bad connections. On my
Compaq (aren't Compaq and HP the same these days?) the socket is notorious
for coming completely loose from the motherboard resulting in the whole
thing failing. My PC was dismantled carefully, the socket soldered back
with some suitable copper wire where the MB contacts had failed and then the
whole socket epoxied to the motherboard for further strenthening. Should it
fail again, I'll never be able to repair it again but frankly I've repaired
it once too many times for my liking already.

Paul DS.