Hi and welcome to our forum! What kind of coach do you have? I don't think you will be able to run an AC unit off of an inverter. They draw a lot of power and although they do make inverters large enough to power one, the battery bank would have to be huge!
We have a 1750 watt inverter that runs out TV, fans, and a few lights when we dry camp. It runs the microwave with no trouble but I do have to watch what else is on line at the same time. I vacuum with it daily. I have two six volt golf cart batteries that are dedicated solely to the inverter and two more golf cart batteries just for the house. That way even if I kill the house batteries, I still have the others or vise-versa. I can tie the two banks together if need be but so far I haven't needed to.
I too plug the main power cord into the inverter and I turn off the charger and set the 'fridge to gas. I have a remote switch in the hall. See my earlier posts for pictures
Hope this helps...
2002 Damon Challenger 348