When the compressor kicks in it will draw considerably more power for a few seconds, to get it up to speed quickly. This will kill your UPS, as noted.
There are UPS systems out there that will automatically shut your system down and then turn everything off. That might be the way to go. But, if you only lose power for a short time, maybe you don't need to worry about the AC. You could use fans to keep the air moving until the power comes back on. You have to determine just how long it's off and how much heat you can tolerate, for that short time span. If you are talking minutes then this will work. But, if you are talking, say, an hour or two, then you'd be better off doing a controlled shutdown, while on the UPS, and waiting until the power comes back on reliably.
At least one of the automatic systems makes an image on the hard drive of everything in memory and shuts down. When the power comes back, it automatically restore everything back where it was. I have seen these work and they are truly amazing, tho not real cheap. But in areas where the power is unreliable they are invaluable.
Another idea is an event recording system that automatically backs up all activity to, say, a DAT recorder. If the system dies for any reason it's all on tape right up to the moment of outage. You might lose the last few keystrokes, but that's it.
I hope this helps.
Bob
A positive attitude won't solve all your problems, but it will annoy just enough people to be worth the effort.