There isn't a "hot wire" into the trailer socket as part of the towbar package (and on a standard 7 pin socket there isn't provision anyway).
A 12 pin socket does provide this facility.
I would suggest to be very careful with simply running an unprotected hot wire from the Jeep battery (no there won't be any canbus issues) to the trailer socket without some means of switching it off when the engine isn't running.
A typical 3 way fridge as used in most run of the mill caravans will flatten a battery in very short order (the heating element in the absorption fridge consumes 12 amps or more). Wire sizes must be suitable for this sort of current and minimise voltage drop.
What vanners do is to install a battery isolation switch - as simple as an ignition switched relay, or a proprietary motion switch, or a specialised battery isolator.
Charging the van battery is another issue and will require heavier cable to the van and more protection to your tug. Anderson plug and sockets are generally used here - a separate ballgame.
You would need to check the Jayco to see what you would need to match. Perhaps ask your hirer for specific requirements (they'll give you recommendations) and mosey over to Caravaners Forum • View forum - TECHNICAL Forum - Caravans, Motorhomes, Vehicles and Towing.
and have a prowl around. What you are looking to do isn't a major issue but will take a bit of preparation of your vehicle.
I'm not aware of any safe way to do a "casual" setup - likely a scenario for something to go wrong and spoil a perfectly good holiday - done right means peace of mind.
Enjoy the vanning - the Jeep is a brilliant tow vehicle.