Is there any way to use a UPS properly when there's no ground wire in the outlet? (Uninterruptible Power Supply)
While a UPS, and many surge protectors as well, will show wiring faults equivalent to a magic 8-ball, a UPS should never actually use ground, except if there is a true fault. A brownout or blackout should trigger the USP battery/inverter but not actually put any current on the ground wire. In fact, if it did then it would immediately trip the GFCI (assuming that some of that current should have been going on neutral).
More likely is a faulty UPS. The most common fault, unless it is brand new, is a battery that no longer holds a charge properly. UPS detects fault, switches to battery/inverter and promptly fails. Most small UPS now have user-replaceable batteries, though whether that makes sense or not depends on the age of the UPS (other parts can fail over time) and whether it might be time to upgrade to a larger UPS for other reasons.