However, It also my understanding that when the caravan is hooked up to the car there is only enough power provided by the alternator to keep a pre-cooled fridge cold enough to store food. There certainly isn't enough power to get a fridge cold from scratch.
That is why most modern caravan type fridges albeit called three way, only work properly on mains electricity or gas. A fridge would flatten a leisure battery in no time at all.
I'm sorry Paullus but that is just an old caravanner's tale. There are reasons why when running in 12V dc the cooling performance of the fridge can be compromised, but it's not from lack of power of the 12V element.
Whichever way you power the fridge, roughly the same amount of energy is consumed by the fridge (Watts per hour , and that means the potential cooling effect is also roughly the same.
The compromises arise, firstly becasue the 12V system only runs when the tow vehicle is providing power which is only when the engine is running, and producing enough excess charge. Depending on the vehicle and the journey that could mean the 12v is only intermittent, which reduce the overall cooling capacity of the fridge.
A second compromise, is the fridge is designed to normally operate when stationary, in virtually still air conditions. The motion of towing can cause unhelpful air movement around the fridge which can compromise the heaters operation.
So its not that 12V is incapable, its other factors indirectly associated with when 12V power is normally used.
Given time in still air 12V operation would be almost indistinguishable from mains or gas.