If you took a Wall Wort transformer say the 12 volt DC Version and find a small 12 volt DC relay, you could wire it into the close cover switch so when you un plug the wall wort it opens the cover switch circuit. Plugged in all is normal....

Now plug it into a outlet and your UPS into into another. Now when your power blinks or the power goes out the UPS takes over just as the cut motor is stopping due to a open cover. You don't loose the project and the UPS has a small load of just the basic electronics of the computer without the high draw of the cut motor.

AL