12v whilst towing

With standard caravan wiring, NO....all the normal domestic caravan 12v ciircuits will be disconnected. (The Fridge and battery charging circuits are fed separately from the car).

There is nothing to stop you adding an additional 12v outlet wired directly to the battery (via a suitable fuse).  As long as your battery IS being charged from the car, then the car should effectively be supplying the power.  Do be careful though!  ...  all too easy to arrive on site with a flat caravan battery!

I would prefer to put the cold-box in the car and power it directly.  Even then be careful if you take long rest stops ....they do draw a good few amps.

We use our cool box in car, and the selected socket in the car switches off when ignition is off, so no risk of flattening car battery. We also add freezer blocks (refrozen in caravan fridge during overnight stops) to add cooling eff3ct.

I think that your essential considerations are these. While towing with the van’s built in fridge set to 12v, it is supplied only from the car, which in turn protects itself by disconnecting that feed when the engine is turned off. All 12v sockets in your van are supplied from your van battery, which should top up from the car while towing, but possibly at a slower rate than it’s discharge from your 12v fridge. The power consumption rate for portable 12v fridges varies in proportion to their effectiveness. A good fridge will draw a lot more than a poor one. As a rule of thumb, it’s best to plug a portable fridge directly into the car, carrying it in the boot. Even then, you need to pay attention to the output rating of your car socket which can easily overload.

We fill up the caravan fridge and freezer so we have stuff for the first few days, and set it to 12v while towing,  then we find a shop to restock.

We also have a 12v/230v coolbox and use it for extra drinks, picnics and  shopping, like others we plug it into a socket in the car that turns off when the ignition is off.

When on site we plug the coolbox into our outside socket if we need to run it.  I would not want to carry it in the van while towing as it would add a lot of weight when filled up.

We have a mains/12v/gas fridge that we use and it travels in the boot and powered by 12v whilst traveling. It’s more readily accessible and can be controlled much easier than if in the van.

We occasionally carry a small electric one in the boot, plugged into the socket in there. I suppose one could be added, or trail it through to the front. This enables us to keep lunch cold and not open the caravan fridge. It then gets used for beers on site.😀  

At jennyc ... All 12v sockets in your van are supplied from your van battery,

Apart from when connected to the towcar with its engine running when the habitation relay switches off power to caravan's 12 volt supplies.