The Check light will come on regardless of AC or NO AC if at any time the refer tried to light on LP. And it did not light. BUT with a check light, the refer should still work on AC if the AC system is operating.
Now, On AC check for 120 volts at the board on the 2 wires coming from the Heat element. IF you have 120 volts, then the Heat element is bad regardless of OHM reading. IF NO 120 volts, odds are the board is bad, if you have 120 volts on the 120 input to the board. Do you have a year model for the refer or RV? What color is the epoxy the resistors and circuits on the control board? Cream colored or Greenish colored? Doug