When you troubleshoot these it is going to be a process of elimination. I am going to give you a service manual which is detailed and will allow you to check some components out. Click HERE for that manual.
IF it is starting on gas and then shutting down go to the outside access panel and find the burner tuber, the burner tube will be below the electrode or what sparks, get an air nozzle and blow out that tube really good, sometimes bugs and debris can get in the tube and cause problems.
Check your DC voltage going into the fridge, the fridge requires DC voltage to turn ON, if the voltage was low the fridge may not work correctly, it should be over 12 volts going into the fridge.
Also if it works on AC power and not LP it could be a bad gas valve, LP pressure problem, circuit board etc, the manual is needed to do some in depth checking..
Please let me know if you need more help on this, respond back to this and I can assist you more if needed....