I don't know how large a refrigerator you need.
But Lowe's has a 4.5 cf compact fridge from Frigidaire, that only draws .88 amps while running and 1.01 amps on start up.
$200
It is almost the same size as the current fridge in my Toy RV.
That should be within range of a couple of good batteries. It wouldn't be running 24 hours steady, just on and off to maintain the temperature.
You could even turn it off at night and still maintain the cold.
A solar panel, running the generator, or RV a few hours a day, should keep the batteries charged.
An Ideal situation would be the solar panels.