Firstly the 12v willonly work when connected to the car and the engine running.
However, as it fires up on gas but stays warm, look at the back of the fridge by removing the vent covers and look for yellow staining around the burner area, if there is staining, the coolant has leaked and the fridge is scrap.
Another test is light on gas for half an hour and then feel the top right of the cooling coil just to the left of the burner chimney, it should be hot.
Now feel the left hand top, it should be almost as hot, now feel half way down the left hand side, it should get gradually cooler towards the bottom,
If it is hot by the chimney but nowhere else, the coil is blocked and the fridge is scrap.