I run a Xantrex Powersource 400 battery backup on my tank. It's a great unit, but unfortunately they don't make it anymore. It'll run a heater for about an hour, at most, but will run the low wattage stuff on the tank for several hours. Most consumer-grade computer UPS units don't have enough power stored to run stuff for very long. Don't even think about running heaters on them (or lighting for that matter), but they *will* power pumps and powerheads to give you some aeration for a limited amount of time. If you wrap the tank in blankets, it'll stay warm for quite a while all by itself.
My main power backup though is a Honda generator. The battery backup just keeps things going during those short outages, or gives me some breathing room before I have to decide to drag out the generator and hook it up.
You really do need to figure out though what is wrong with the circuit. If your GFCI is tripping for no apparent reason, having a battery backup doesn't solve that problem. In fact... here's a scenario that is kind of scary: you have a short somewhere in the system and just as you're getting shocked, your GFCI does its job and shuts things down. But then... your UPS senses a power outage and kicks in, sending power back to whatever was shorting (and shocking YOU) in the first place. And now since the power is coming from the UPS and not your house wiring, you have nothing to protect you except for the built in circuit breaker in the UPS. For that reason, I always run a GFCI on the output side of my UPS.