If your batteries were dead the refer needs 12v for control board to operate. You should have had 12v's from your converter supplied to control board while plugged into shore power. Your converter also charges your house batteries. I would check the converter for some problem make sure ckt braker is operated,power at outlet for it and if not maybe on GFI ckt supplying outlet.
You can also check the burner tube and make sure no cob webs in it and the auto spark points near end of burner tube are clean and about 1/8" gap for spark. If points are not good you can change for better spark.
You should also check the age of your batteries they maybe to old to hold a charge.