You could consider one of THESE.
I used one in both my Pop Ups and was quite happy with it. It works like an icebox when you have no power (Tho it WILL run on 12v, it isn't very efficient) and it has a 110 adapter for shore power. I never actually checked the 110 via invertor, but I bet it is more miserly than a compressor fridge
is. (Just a guess...)
Use the Blue Ice when you are in Ice Box mode, it will last for days in just about any condition, sans the desert.
No freezer tho.
My link there is NOT the cheapest I have seen them. Wally World has them for 65 bones.