I vote for the charge line, as already suggested several times. 'Tis the easiest and best solution, IMO. I had to wire one to our Ford Fusion (hybrid) that we tow, and it works great. Very easy, quick, and cheap to install, too.
If running a charge line is not an option, then yes, you can idle your Jeep's engine to prevent the battery from draining. However, for reasons already mentioned, I would NOT make a habit out of doing that. Once or twice as a temporary solution until you could put together something better, maybe, but not for a permanent solution.
One thing nobody has mentioned, that is my biggest concern with the idling approach: You can't see the dashboard of your Jeep, so you have no idea whats going on with that idling engine. If it stalls, starts to overheat, or has any other issue you'd see on the dashboard, you are completely blind to it. If I was going to do this on a regular basis, I'd look at wiring up a wireless camera of some kind in the Jeep, point it at the dashboard, and mount the monitor screen up in the cab of your MH. That way, you could keep an eye on the Jeep's dashboard somewhat.