Frigidaire makes some small window units 5k-5.2k BTU that have what they call, "low-voltage startup". It's just a delay timer so that the fan comes on first and the compressor a few seconds later, this helps keep the starting surge down.
Some years back I asked Frigidaire tech support about running those units from a generator and they sent me detailed specs on the 5.2k BTU unit. It had a SEER of (IIRC) 11 and the startup max draw was 1300 watts. That unit could be run from a small "2000 watt" generator (almost all "2000 watt" gens are really 1600w gens that can handle a momentary surge load of 2000w - the Honda eu2000i can handle 2000w for up to 30 minutes).
But yea...you won't run any a/c unit (or electric heat either) for very long on a small battery bank. Figure if you had a 105 amp*hour battery at 12v:
105a x 12v = 1260 watts
So if the a/c unit drew 600w running, then you'd get maybe 2 hours before the battery w as totally flat.
This guy runs a standard 120v 13.5k BTU overhead RV a/c unit from a Yamaha ef2400is generator no problem:
http://www.2manytoyz.com/yamaha2400.html
EDIT: I forget the name of the guy - the Canadian guy who bolted a Bigfoot trailer on a Fuso and drove it around the world. I remember in one of the pics I saw, he had a small window a/c unit installed under a bench seat in the camper and a Honda eu2000i.
I would bet that most 5k BTU window units could be run from the Honda.