Now I'm running a Dometic CFX 40. Before that I had a CF 25, which only ran on 12 volt. So when shore power was available I would put a charger on the battery, which was able to fully charge the FLA battery while keeping the fridge running, so I understand what you're doing.
I would recommend that if you're planning to run an A/C, the converter and the battery charger simultaneously you consider using a detachable 10 gauge 30 amp RV service with at least 2 15 amp breakers. 15/30 amp adapters aren't that expensive and you can use one at home when you're only running the converter and charger.
That's the future plans for it. I couldn't agree more. I did it this way just to be able to run the fridge 24/7 without draining the battery and I already had everything I needed except the 115V plug I mounted on the topper. I am doing this mostly as proof of concept for the wife and to see if she and I can be comfortable sleeping in the bed of the truck. It is working out great so i want to go back and re-do some wiring to make everything code compliant and I have already created a list of things I need, including the 30A cord and a 2 space, 2 circuit breaker panel.
The AC is a residential window unit that operates at 450W, so only 3.75A at 115VAC is needed to power it. The fridge barely uses 50W when running so that is only 4A on 12VDC. If I am at a campground, I can plug the AC directly into the 115V there and bypass the converter altogether to reduce the load on the converter wiring, for now. My 2500W Westinghouse generator has two 20A (5-20R) receptacles on it so I can put the AC on a separate circuit when I am camped remotely.
I appreciate the advice!