I would suggest that you check the Wiring at the Rear of the Jeep with the 7 Pin unplugged. Check my attachment to find Terminals #1 & #4 with a Volt Meter or even easier with a 12 Volt
Light Bulb with short pigtails (easy to make and fairly inexpensive but very valuable). You most likely have 12 Volts there even with the Ignition Off.
Agreed to be sure that the 12 Volts Power source should have a Fuse or Circuit Breaker at the beginning of the circuit (within a few inches of the battery or terminal block). I would suggest that you NOT use a self resetting circuit breaker for obvious reasons.
Charging Systems are very different between cars and say boats. Cars are designed to recharge rapidly and then drop off shortly afterwards because of the fast discharge of the starting loads. Other Charging Systems are designed to charge at higher limits for extended periods of time making them much more efficient although at a much higher price. The alternator and wiring has to be much different to make it last with these higher outputs.
I am going to use a Regulator for the incoming power and before it gets to the battery or batteries (I am still in the physical restoration stages). This will ensure that I am not overcharging the batteries in the AS. The one that is at the top of the list right now is here;
Toad-Charge Towed Vehicle Battery Charger & Maintainer Kit - RV Power Supply
Do some measurements but the 40' version should work for most of us. This is designed to work with a motorhome and the toad (or towed vehicle) and that is how I found it but it (sure looks like what I want in my AS). I am setting up a 2005 Wrangler Rubicon to be towed by our 1981 Newell Motorhome (yes another riveted aluminum coach).
Trailer Wiring Diagrams | etrailer.com