12v dc plug not working

I assume you mean the 12v socket on the fridge?
The end of the lead has the plug.

Looking at a manual for a primus fridge, it is recommended that a fuse be included in the circuit to the fridge. This may indicate there is no internal fuse on the fridge itself.
Take a look at the area adjacent to the 12v socket. Can you see a bladed fuse of some kind?

Does it run OK on 240v?

What voltage measurement are you getting at the end of the 12v lead plug?
The fridge appears to have 3 different settings for low voltage cutout.
The voltage you are getting should be 12.8 volts or above, which indicates a full battery capacity. Any voltage drop in the circuit may reduce the voltage below the minimum necessary to overcome the low voltage cutout device.

The low voltage cutout device itself may also be faulty.

The chances are you need an expert to look at and test the fridge operation both on 240vac and 12vdc and diagnose the problem.

check all your connections back to the battery , you can have 12v at the plug end with no load ,
but a poor connection between the battery an fridge somewhere it drop off under load
an ya fridge won,t go .
been there an done that
mechpete

Is this a 3 way rather than compressor fridge? If so the 12V heating element may be cactus.

See if it works with the engine running, a lot of the the time the extra voltage will overcome poor wiring. If it works like that then you may need to check the car wiring. If not you may have a fridge problem.

Wiring for fridges is always a pain if its not quite right!!!