Jeff Liebermann says...

The relay arrived this morning, and I've installed it,
and everything works fine so far.

Congrats, so far.

I have to keep the old relay in case I ever need to
send the unit in under warranty. So I won't be taking
it apart. But, it does rattle when I shake it, which I
think is not a good sign.

It might be a spring. Some DPDT relays have two
springs, one of which might have come loose. With a
spring missing, the tension on the normally closed
contacts will be reduced, possibly causing the symptoms.

Well I'm going to declare victory and withdraw from the
field of battle.

It's been a week now, and the UPS has passed the boot test
every morning without fail. And as luck would have it, in
our 100-degree weather there have been two short-term power
outages (two that I know of because I was at the computer
each time) and the UPS switched into backup mode perfectly
each time.

So it appears the problem definitely was the relay, and my
replacement operation was successful. And all for less than
the cost of shipping the UPS back to APC. Plus, if
something goes wrong in the future, I'll know what to do.

Thanks very much for your help, Jeff.