I would like to use a Raspberry Pi with both the Multi-IO HAT and the Multichemistry Watchdog UPS HAT in the same stack.
Both boards seem to accept 10–28 V input and both appear able to power the Raspberry Pi.
I plan to use a single 12 V / 2 A power supply.
Could you please confirm the correct powering method for this setup?
Should I power only one board, and if so, which one?
Or can both boards be supplied from the same 12 V source without any conflict?
You need to power both cards from the same power source, no electrical conflict here.
The only thing you need to take care of is, in case you need to use the watchdog function (repower in case of Raspberry Pi hung), you need to synchronize the two cards so both will cut the Raspberry Pi power at the same time. If you need only the battery backup, don't worry about that.
One more thing that needs to be mentioned is that the Multichemestry Watchdog will back up only the 5V rail.