If it "runs and gets hot then shuts off after about 5 minutes and won't relight" I'd suspect the thermostat/limit switch. There are two, one for 120 VAC operation of the electric element and the other for 12VDC operation of the LPG burner/system. I'd change out the 12VDC thermostat/limit switch. If that doesn't fix it, next is to replace the control board.
It's apparent that the ignition system is operational, the water heater lights, runs for a while then stops, so it's probably something to do with the temperature control part of the water heater. When it gets to a certain temperature, it shuts off (probably working as it should) but then as the water cools, it doesn't relight, leading me to think it's a thermostat problem, since it will "relight, but goes out again.
2015 F250 6.7l 4x4
2014 Cougar X Lite 27RKS