Simple test (Fluke meter required). Plug a 400W+ inverter into a battery and turn it on. Place the Fluke meter on the positive lead coming off the battery and switch the meter to read amps. Plug the fridge into the inverter and switch the fridge on. If the amps rise, the fridge is drawing power, perhaps the heating element has shifted? That would be something to pay immediate attention to. 240V element is separate to 12V and of course is separate to the burner nozzle.

Personally don't run mine on 12V, because 12V doesn't use thermostat - it draws constantly and cools at a "maintenance" level. Both 240V and gas use thermostat and will switch on and off as needed. When driving, I invert the incoming 12V and have the fridge plugged into the inverter.