Does the fridge really need to be cold upon arrival?
Get the fridge powered up on arrival, nip to supermarket for supplies, it should be cool by the time you're back.
This caravanning malarkey seems a hell of a lot more effort than camping, for the gain of being more cramped, restricted on travel speed/distance, drink huge amounts of fuel, and storaing the bloody thing
Fridge is not really a problem, we get it cold before we set off, and will probably get the odd current from the alternator on the journey.
The main problem would be the caravan battery draining to the same voltage as the car battery.
I think the simple solution would be to fit a battery isolator switch (caravan battery end) to stop this happening as you don't need the 12v system on the caravan whilst towing.
Err cramped? Don't think so, fixed bed, all mod cons, hot & cold water on tap, waste straight into drain, central heating if it gets cold, own spacious shower, loads of storage space, never gets damp in the pouring rain, I could go on............