Being hooked up to shore power may not necessarily work especially if you have a bad battery.
Shore Power provides only 110-120 volts, but there is a converter in the rig that if working correctly converts incoming 110-120 volts to 12v for things such as a 12v furnace. When on shore power the furnace still operates on 12 volts via the converter AND the battery! If you have bad battery it can affect the converted power, so if you know the battery is bad, DISCONNECT it so that it doesn't interfere with what the converter is trying to do. Besides the converter providing most 12v power needs it's also trying to recharge the battery or in your case, probably a 'bad' battery and this can cause problem.
I have dual 12v batteries and one of them was bad and caused very strange 12v supply problems (slideouts not going in or out smoothly, dimming 12v lights when turning them on and off, and other problems) even when connected to shore power. When I found out one of the batteries was bad and removed it, these 12v problems went away.
Since your ignitor sounds like it's clicking away, this might not be the problem but you might try disconnecting the battery and take it out as a possible problem cause.
You might also have a bad regulator, but with the other propane appliances working okay, I'd suspect something is not right with the furnace. Like maybe the thermocouple. Isn't that the thing that keeps the propane flowing when the control board tells the furnace to start up? The clicking means the control board is trying to tell the furnace to start up, but if propane isn't flowing it could be no propane, not enough pressure or something not letting the propane through like a blockage or more lightly a bad thermocouple. This happened to me on an oven once. Same principal.
Any furnace guys out there?