I'm not completely clear on which of your batteries you are going to use to run the inverter, but in either case you would want the converter disabled so you are not trying to charge a battery with a battery. You will get the convenience of all the camper's AC outlets being live, just like if you were plugged into true shore power. But that doesn't mean it can operate just like when plugged into true shore power. A single (or even both 12V batteries working together in parallel) will not be able to operate much. Small TV no problem, mini fridge maybe for a short while, AC unit no chance, space heater no chance. The solar will not increase the capacity to operate things really at all, but it will provide a little charging when there are no other electric loads in use.
This.
You will need more or different power if you want to operate even minimal amp draw things for any amount of time. Microwave also will not work on one battery. If you want to go batteries, then you need more of them, and a LOT more solar or a way to charge them. Even then you are pretty limited, and you won't be able to run the AC and probably not a heater and will use up most of your amp hours with a microwave even on short bursts. This is a math problem. You have a certain amount of amp hours you need. Microwave takes X amps to run at Y length and frequency. How many amps does your microwave require? Add up the amount in a day you require it and total it. Same with your fridge, etc etc. Sum all the amps required for everything you plan on running and that is your total required amps to support usage. Then see how many batteries you need to support that demand. Then you have to consider the charge rate to keep your batts working while drawing a charge. I think you'll quickly realize that unless you are extremely frugal even with a large battery bank and a lot of solar, it will be challenging. By contrast a generator (which wouldn't really be more than a good solar set up - if that) is pretty endless with a power supply sans the fuel required to keep it running. Generator hours at any campground needs to be applied here. Another option is a cheap campground. Some state parks are as low as $25/night with electricity. Usually state parks don't have water/sewer but dump stations are an easy remedy. Geography is also a consideration. In the northern half of the US most campgrounds close in the winter.
So, like I said before. Lots of factors. You need to know your need, what you need to support that need and the most cost-effective way to get there.
Not saying it can't be done, but it's a math problem. And as with most things, usually throwing money at will solve most of those problems. Rving can be cheaper than conventional living but it has to be set up right. But it's not always cheaper.
