Now that you've shown a bit more of the circuit board, I can see that it's encased in a black plastic box so that the back won't be visible.
The heating element, K in the diagram, will be around 240 Ohms. The circuit board will switch it on and off depending on the temperature. The wiring to the heating element will have much less resistance, as it shouldn't get hot.
If your fridge works fine on gas and 12 V, my suggestion would be to put the mains directly onto the mains heating element. To maintain the temperature control, you would need to fiddle a few things, so here is what I think you would need to do:-
1) Find a 230 Vac input 12 V power supply. It won't need to be a high rating, and 1 A will be plenty. You've probably got a few from old routers etc. Connect that so that it is powered by the mains input. A power supply with a wired input or a
C8 input might be more convenient to wire up than a wall wart.
2) Connect a change-over relay so that the 12 V of the fridge is powered from the power supply when the power supply output works, and from the normal 12 V feed when the power supply doesn't work. (You can use an automotive change-over relay for that. Wire the negative of the power supply to the negative of the normal 12 V feed and to pin 85 of the relay. The normal supply goes to pin 87A and the output of the 12 V power supply goes to both pins 86 and 87. The 12 V input of the fridge is powered from pin 30. You need heavy-duty wire for the normal supply and the connection to pin 30)
3) Connect another change-over relay so that the fridge's 12 V heater can't work if the mains is on. (You can do that with a second change-over relay. Pins 85 and 86 are connected as the first one. Disconnect the wire from pin 87 of relay "O" on the diagram and connect that to pin 87A on the second change-over relay. Connect pin 30 of the second change-over relay to pin 87 of relay "O", also with heavy wire. Do not connect anything to terminal 87 of the second change-over relay)
4). Get the fridge circuit board to control the 230 V heater. Find a normally open relay, 12 V dc coil, mains rated contacts, so you can't use an automotive relay for that. The contacts only needs to be rated to 2A or so. One like the relay in you first photo would be fine. Connect the coil in parallel with the coil of relay "O". Wire the the live mains input through the contacts of the relay to one side of the 230 V heater and connect the other side of the heater to neutral.
Do not connect the mains, live or neutral, to the fridge electronics.
The plan is that the fridge electronics will run from 12 V, and behave like it is on 12 V, when in fact the heater is running on 230 V ac. When 230 V ac is connected, the fridge electronics will be powered from the power supply. The 12 V heater will be disconnected to stop it overloading the power supply. When the fridge electronics turns on the 12 V relay to power the 12 V heater, the relay for the 230 v heater will turn on, so the 230 V heater will run.
When the mains is disconnected, the fridge electronics works as normal. If it runs on 12 V, the relay for the 230 v heater will turn on as well, but it will have no effect because the mains is disconnected.
The plan won't work well if the fridge electronics has a very different strategy on 12 V that is has on 230 V ac.
I assume the mains input will be fused at 3 A or less. The plug fuse will be fine for that.
There is a lot there that can go wrong, so draw it out, and post the drawing here if you at all unsure. Be very careful fault-finding with any 230 V stuff. What I have suggested minimises the mains wiring and all the relays will have 12 V coils. The heater wiring for the 230 V stuff must be properly insulated and kept clear of everything else. The heater wiring for the 12 V stuff must make good connection as it is high current.