My electrolux fridge was 3 way powered, ac, gas or 12v, the thermostat only worked on gas or ac, on 12v the fridge just cooled constantly and sapped a lot of power, around 100w, if left a while everything froze up.
I've seen boats that have a battery just for the fridge.
You can get cold boxes these days that use peltier heat pumps and are a lot more efficient, and not expensive either.
Like I was saying before you make any changes I'd suggest you do a power survey, work out exactly what you need, in my experience this is rarely done.