tsanjose,
No one runs an A/C from battery. Not even RoyB. He has a battery bank yielding a total of 255 amp hours. Taking the rule of not dropping below 50% capacity, that leaves 127.5 amp hours.
A typical (Coleman Mach 3) 13.5k BTU air condition runs at 15.3 amps on 120v AC. Using an AC Amp to DC Amps conversion that incorporates the inefficiencies of an inverter (I used
THIS site), you end up with a pretty hefty draw on your batteries.
At 168.9 DC amps required to run the air conditioner, RoyB's set up would effectively power the air conditioner for 45 minutes before needing to be shut down and prevent overly discharging the batteries.
There may be some rough calculations and things not taken into account, but you get the idea. The calculations do not even take into account the start-up amps required to get the fan and condenser running.
It's not going to last overnight. A generator would be needed to run an air conditioner.