I guess you could run the AC with your batteries charged by solar but your asking a lot from your panels and batteries. Like the previous poster has said do the numbers first.
As far as keeping your batteries from being charged from the converter/inverter just flip the converter breaker off. Just be sure that the converter is the only thing on that breaker. I had a some outlets on that breaker so I added another circuit breaker for them. I will be adding a relay to automate that procedure soon. So when it senses my inverter being on it will drop that panel breaker.
Ron W.