This one is OK. It automatically switchs between shore power and battery power when you plug in.
https://www.amazon.com/Power-Inverte.../dp/B00TI1D5JK
You do need to know a bit about 120 volt wiring to install it.
With all inverters, its important to keep it close to the batteries and use proper size cables. The 120 volt wires can be longer with out a problem.
If your not comfortable wiring 120 volts, I recommend the Xantrex ProWatt 1000. With that, you would need to swap the fridge plug between it and an outlet, once plugged in, or just leave it running on the inverter full time. The converter will keep the batteries charged anyway.
I don't think the residential fridge in my boat has ever seen shore power and its 9 years old. Its plugged into the inverter full time.
As far as batteries, If your plugging in while camping anyway, just use what you have. You should have a charging line from your tow vehicle to keep them charged while traveling. If they don't last on a trip add some of the same type to them.
You don't need " true " deep cycle or " AGM " batteries unless your boondocking.