What do you mean by "the gas won't turn on"? Do you mean the burner won't startt, or it won't stay lit, or it tries to light but there is no gas there at all?
If there is no gas at the heater, try lighting the stove. If it lights and burns steadily, the overall gas system should be ok, and I'm out of my depth. Based on what is actually happening, probably someone out there can give you good advice, but they need more data to do so.
When I say the gas won't turn on, I mean the gas valve does not engage. As the blower motor reaches 75% of the normal rpm, the combustion air switch, in response to the air flow, will engage allowing current flow to the gas valve. The current to the gas valve opens it and allows gas to flow to the burner.
This is how the heater is supposed to work. I have the blower working, the ignitor working, but no gas when on 12v. When using 110v, it works fine.
Have you checked to see what the actual voltage is to the motor of the furnace blower when shore power is not connected? Since the only difference to the furnace between shore power and no shore power is the source of the 12V power supply, I am betting that you have a weak battery that does not supply enough power to the motor to turn the blower fast enough to close the sail switch. It takes a minimum of 11V at the motor to turn it fast enough to close the sail switch. The purpose of the sail switch is to prevent the opening of the gas valve when the blower is not up to speed to provide a safe air movement. The voltage supply to a DC motor is what determines the speed of that motor. You can verify that by checking first the voltage to the furnace and also check for 12V power across the pins of the gas valve. I strongly suspect that you will not have power to the gas valve.