My key was exhibiting the exact same problem approximately a year ago.
There were times the engine would start and run; after several minutes it would die.
Other times the engine would start and run fine.
After testing numerous electrical connections and reading voltages and resistance on relays; I noticed the amber SKIM key symbol was flashing when the engine died.
To test a theory I brought a spare key with me during a drive when the engine died, tried cranking the engine on and off for several minutes and it wouldn't start. Swapped to the spare key and the engine started immediately and ran fine; switched the engine off and tried the original key and the engine would start.
Since resolving that problem I have labeled the problem key "Bad Chip" kept since the key cuts are in very good condition.
NOTE: There was a series of SKIM keys that were produced with bad chips; I believe it was the C series.
A friend of mine worked at the Jeep dealerships for many years found this out the hard way while programming keys for my Jeep.
The dealership returned ALL keys of that series when he brought it to their attention that the keys were produced with faulty chips.