Another thing to check, it might sound stupid, but have the battery load tested.
The Jeep can throw all sorts of weird symptoms, when the voltage drops on start up and it is worthwhile checking the battery.
It is just you said, if you start it and run it, then switch it off and start again there is no problems. This could be because it takes a lot less current to start a warm motor, than a cold one, so you will have less volt drop on start up.
Anyway it is a no cost check worth doing IMO.