I am assuming you want the lowest cost solution. You can go with no house batteries at all. That implies the fridge and other AC loads are off when traveling. The next step up is two Sam's Club 6V Golf Cart type batteries. You will need to learn to maintain them properly, keep them watered and charged, check specific gravity, etc.
Then you have to size your inverter to match the load. Minimal solution would be to just support the fridge and some minimal loads like a laptop charger. You can probably get by with a modified sine wave inverter which will save some money as well. I can't guarantee it will work for everything though, that is why I spent the money for a pure sine wave inverter.
You can forget about running your Air Conditioner (AC) off of the inverter, or a microwave, or a toaster, or even a coffee maker, unless you get a big inverter, and then you can forget about running the AC off the inverter unless you have substantially more batteries. Running any of those things implies firing up the genny or plugging into shore power.
For a small AC the Coleman polar cub is one of the lowest current draw units. Insulation is great, but it still takes quite a while to cool down the thermal mass inside the insulation. You can probably expect the AC to run on the compressor 50% of the time if you are in 105 degree heat, and that is only after it stabalizes at the desired set point. Unless they are double glazed, windows are terrible for letting heat in/out, even double glazed not so great.
There is no substitute for doing the math to figure out what you really need.
There are lots of ways to connect between the starter battery and the house batteries, including no connection at all, which isn't a bad idea. Some use a latching relay that only connects when the truck is running. Some use a pair of diodes so one battery can't draw down the other. Some use a fancy (and expensive) solid state device that is smart about making the connection. I used a simple and cheap 200A battery disconnect switch from the auto parts store. I use it rarely.
If you get into solar pannels and all that then you are way beyond a minimal solution. I can't see how it works for your situation anyway, but it's your call. There are some regulars on the Escapees forum who are much more qualified (than me) to give advice if you decide to go that way.
If you have batteries you need a battery charger. I would suggest one that is integrated with the inverter. You probably also get a transfer switch in the bargain. One advantage of buying the package instead of piece meal is you know that it will all work together. Ideally you want a 3 or 4 stage charger that will pump up the batteries quickly so you don't have to run the genny for a long time. If you are on shore power this doesn't matter so much.
Many would recommend a battery monitoring system that measures the current in and out, and works like a gas guage. For a minimal system you can live without it if you are prepared to monitor battery voltage and be aware of all the other clues that your systems will give you. One advantage of cheap and minimal batteries is that your learning mistakes won't cost you so much.
You really want a proper roof vent fan, like Fantastic Fan or similar. Lots of times you need fresh air as much as anything, and it is much less of an energy hog than AC.
Good Luck!
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