Tom,
No your quest is not far fetched.
Here's my suggestion(s):
Replace the UPS battery pack with a much larger deep cycle setup. You'll have to use your own judgement as to size and type. I would start off with flooded batteries because they are much more forgiving than other types and are usually cheaper. Buy more storage than you think you'll need. It's always better to have too much than not enough, to me. Solder on much larger (gauge) DC cables to the UPS inverter section and bypass any clamping system for the battery cables. Lowering these resistance points will help the whole setup work better.
Hack the UPS charging circuit to switch on another charger-system or remove it and bypass that weak charger comletely. Or, use a charge controller and PV or wind or wahtever to recharge those batteries. Or keep an AC charger (wall) plugged in at all times, one that will provide some sort of controlled current to protect the batteries. or build a multi input control system, which isn't that tough. The bottom line is that built in UPS charger is a joke but nobody laughs.
I would use the highest AC wattage UPS you can/have. The surge of the compressor cycling on can really suck the amps for a few seconds. Keep the AC~ supply line(s) to the air cond unit as short as possible. I found that my "Portawattz" 1000 will run my little window unit if the feed AC~ line is very short and very thick.
Figure out a way to add more cooling air flow over the inverters heat sink area. A simple setup would be to use a thermistor and a op amp, with the thermistor intimately contacted to the heat sink(s). A real beefy op amp might be able to sink the load of a computer cooling fan or 2, but even a small mosFET would handle that easily.
The UPS may have a higher low battery shutoff point than a regular inverter. I have a small UPS (250 watts) and it will shut down a full volt and a half higher than my Cobra 2500. As far as I can tell this point is hard wired into the design, as I have not found a pot to adjust this in that little unit. That may not sound too important but it's something to keep in mind. This might be a good thing since the batteries will be protected from deeper discharge when the sun sets.
I've found that a small assisting fan, like a microwave squirrel cage fan, which helps move the air out of the AC unit make a big difference. I built a little hanger in front of the cool air port and use one of thise little fans to move the air out into the room with more force. I have no tested data to support this, but it seems to speed up the overall room cooling. The tiny fan doesn't draw much power either.
If you could figure out a way to reverse the UPS' main function you'd have something! If the UPS would stop using the batteries and switch over to grid power when the battery low voltage set point was reached it would be way cool (not a pun). Like this; The sun goes down, it's still hot out and the AC unit still working to cool the house/room. The batteries voltage starts to drop, they reach the cut off point, and then the UPS switches off completely and grid power is passed to the window units so the heat doesn't build. I sure this could be done with a contactor powered by the UPS itself. When the UPS can pull in the contactor, or was making AC~ power, the window unit runs from the UPS-battery pack. When the UPS drops out the contactor is released and the grid power is passed to the window unit.
Keep at it. This ain't rocket science. Above all keep some cool drinks on hand!:-)
AS always, hope this helps.