What voltage is you car battery at after the car has been switched off overnight?
As Albee says, signs of heat damage raises warning bells to bad connections. Check the connections in the caravan plug as mine had severely corroded and I had to replace it, whilst doing this I had to replace the cable as I could not get back to good wire, 1m of wire showed sign of water ingress as the strands were black.
I would put the fridge onto 12v and try taking voltages at car battery and fridge connector (you may be able to get to it via the vents or it may be in a cupboard) with both the car running and the car switched off. I would then repeat this with the caravan battery disconnected.
This should show you if the fridge is getting more power from the caravan battery than the car circuit. Ideally you could measure the current but you would need a DC clamp meter as the currents would normally be above most multimeters.
Haven't tested the voltage across the car battery today, but it isn't connected to the caravan and started fine first thing this morning, so no worries there.
The heat damage is mostly across the 15amp fuse for the battery, looks like a large current has passed through it for a while. I'll replace the fuses and keep an eye on them.
I moved the van on the motor mover while connected to the 240v connector as the van was stranded on the road and the battery was flat, this would, I believe cause some heat damage to the fuse as the ampage passing through must have been right on the limit for a few minutes. I know this is less than ideal but I was in a very sticky situation.
I replaced both of the caravan's connections when I fitted the car's towball as they were a real mess. I'll do a continuity test to make sure there's no breakdown anywhere and I have good core continuity.
I was testing from inside the van, I was putting my probes into the back of the connectors. Got different voltages for the permanent live and switched live from the car (the connector at the end of the grey cable). The voltage dropped on the switched live from 13odd volts to just over 12odd volts when the fridge was switched off and on, so I believe the alternator is doing a good job of powering the fridge. I fried the 10amp side of my multimeter ages ago so only have the milliamp setting but I guess I'd fry that too! (The 'fuses' in the multimeter must have been big as they haven't blown and the circuit board of the multimeter is cooked, buy cheap eh?!)
I've been testing the current draw by removing the fuse and using my multimeter in circuit in place of the fuse, I realise this gives no fuse protection, but the fuses haven't blown under normal operation so I'm comfortable with this.
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