What's the maximum wattage air conditioner I can run for 8 hours through a 12V-120V inverter from 200Ah batteries?
Theoretically, amp-hours times volts gives you watt-hours. Then watt-hours divided by hours gives you watts.

In this case, 12V x 200Ah / 8h = 300W.

The inverter will use some energy, so multiply that again by the efficiency of the inverter. Say it's 85%, then you can run a (300W x 0.85) = 255W air conditioner for 8 hours. Theoretically.

If you can't find an air conditioner with a low enough wattage, you might be able to run a more powerful one at a low power setting, or only run it for part of the time (for example, a 380W air conditioner on for 10 minutes and off for 5 minutes gives a 254W average).