Jeff Liebermann says...

I can hear the relay click when it goes into backup,
and again when it reverts to mains, and I can hear that
whether the backup power gets through or not.

Well, if you're ambitious, open the box and look inside.
If the relay is an open frame type (fairly common), you
should be able to inspect the contacts. If the relay
has a cover, it can usually be removed to reveal the
contacts. My guess(tm) is that you'll find pitted
contacts or a mangled relay armature.

Thanks very much for your reply, Jeff.

The relay is a PCB style DPDT:

Goodsky
EMI-SH-212DCF
12V coil
8 amp contact rating at 250V

I found the datasheet for this series relay, but not the
exact "DCF" part, at:

http://www.goodsky.com.tw/en

searching on the part number EMI-2P over on the left.

But, I haven't been able to find anything equivalent at
Mouser.

The relay is encased in plastic, so I doubt I can get
inside, but I haven't desoldered it yet.

I found 10 with a similar number for sale on Ebay for $79,
but no singles.

This is a really dinky little relay to be rated at 8 amps,
which is probably why it is failing.

If I could find a replacement relay, it would probably cost
less to buy that than to ship the UPS back to APC - it
weighs a ton.

Of course while it would be nice, it wouldn't absolutely
have to physically fit - I could bridge from the PCB holes
to the relay with wire - but the coil and the rating would
have to be right, as well as the speed I assume.

Well, let me know if any solutions come to mind. Thanks
again for your help.