Running a/c from batteries is doable, for a while. Microwaves and blenders do quite well on inverters. TV too.
I heartily recommend upping your system voltage to at least 24 for running air. At 12v it's just too much amperage for your wiring and connectors, and things will heat up.
Many talk about running a/c from a generator, buy in my experience, it's rare. People usually just leave for the day and let the trailer get hot. I myself do not. I run a couple 2200s 8, 10 hours a day. That about 15 gallons of gas every 4 days.