I have a 2500 Onan in my truck camper. It will run the AC but sometimes it will stop, especially if it is realy hot. Aparently, the warmer it is, the more power it takes to restart the air conditionning motor.
I doubt the Honda 2000 will run the AC. If all you want is power to run TV and recharge batteries, the 1000 Honda will do, it is cheap on fuel and very quiet. I had a Yamaha 1000 for use in a cottage to run TV and batteries, it was not running good in cold temp and it finally blew up. I heard that the Yamaha 2500 would run the AC but I would try it first in very hot temp.