I think you're overcomplicating/overthinking this issue.
I have to ask.... when you measure across the terminals of the trailer battery
with the jeep running, it's 12.7 volts? What is the voltage there with the car shut off? I suspect it'll be 12.7.
Here's a quick test that I'd do : just quickly unplug the trailer from the car, and test the voltage at the trailer charge pin of your car end connector. Put your positive voltmeter lead on the charge wire, and the negative lead to the ground pin.
Based on what you're saying in your initial post, I'm guessing that there's no power being supplied at the back of our jeep, and that you need to trace the circuit for an open, blown fuse, etc in the car, not the trailer.
I suspect this because your convertor works when youp lug in to 120. The convertor isn't used when charging from the car, just a simple wire.
Some of the other guys have suggested a bigger wire, which is a good idea, but you should still have about 13.5 volts to the back, even with a small wire. It just might take all day to charge!
The 27 ohms from the trailer connector to the inverter sounds odd, but should only affect the use of the inverter, not the charging of the battery
. What's important is the resistance from the trailer connector charge pin to the positive battery terminal. If that's at 0 ohms, then that reinforces that the fault is in your jeep.