On my next build I am hoping to run a small a/c unit from a 800 watt generator. In theory it should work. The generator will be only hooked up to a battery charger, perhaps 40 amp. The a/c and every other 120 volt appliance will be run from a large inverter. These small generators can run for most of a hot day on a gallon of gas. The A/C will use more amps than the generator will output, (when it starts), but the a/c will not be running continually, I hope with enough insulation it will run for less than half the time. When it is off, the charger will replenish what was taken out of the battery.
If I don't need the air conditioner, running the generator for an hour a day may be enough to keep the battery happy. Without the air running, a solar cell or two may be enough, and then the generator will not be needed to run.