That is the subject of many heated discussions. ON or OFF propane when towing. There are 2 schools. My fridge is on propane when I tow, and it stays very cold.
Many leave their fridge on AUTO, like I do.
That means: if it find AC (120V) it runs on electricity, and falls back to Propane when disconnected.
I do turn it off, prior to fueling the tow vehicle, but admittedly I will sometimes forget.
12V would do a poor job with our larger fridges.
My humble opinion: leave it on if you travel long distances and in warm weather, do turn it off when refueling.
Denis & Diane
* Happy Campers *