In the 390BH, only the rear AC unit is a heat pump. The front one should not have come on.
On the furnace, it could be a couple things. The furnace will not run if gas pressure is low. I've had ours stop working due to low propane but the stove and refrigerator were still able to operate. I'd check your propane levels first.
The order of operation for the furnace start is as follows:
1..Thermostat calls for heat
3..After 15 seconds spark fires
4..gas valve opens
5..Sparking occurs for 7 seconds
At step 2, there is a sail switch that makes sure air is flowing through the system. If the sail switch is obstructed, you won't get any further and the fan will eventually shut down.
There is also a limit switch that makes sure the unit isn't already hot from running earlier. The limit switch prevents the heat of the unit from getting too hot.
If both these switches trip properly, you move to step 3.
If you are hearing the clicking noise, then you've got a spark and you're at step 4 then immediately at step 5.
If you are in fact making it to step 4 or 5, it could be the gas solenoid valve isn't opening. It could also be that the gas pressure is too low, so again, check the propane levels first. Check them at the tank. Don't rely on the inside panel and lights.
Check to see if voltage is coming out of module board to gas valve after the 12 - 18 second delay. Check the wires in the molex connector to be sure they're intact and making contact with the module board. Check wire from the module board to valve for continuity. Wire and connections check OK -replace module board.
If Voltage is coming out of module board to gas valve, but gas valve does not open - replace gas valve.
If you find you aren't making it to step 4, then it is probably the sail switch. It sits by the Squirrel cage fan and senses that air is flowing through the system. I had my sail switch get stuck and even though air was flowing, the furnace wouldn't attempt to light.
It isn't too hard to remove the furnace to do diagnostics. It's a self contained unit with pipes that fit inside the ports to the outside. You can remove the screen under the fridge, loosen some bolts and slide the unit into the RV enough to work on it. You don't need to remove the gas line if you're careful.
Hope this helps.