Was not aware that Dometic made furnaces but speaking for the Suburban furnaces used in most RVs there are several fail safe features that would keep them from lighting.
When the thermostat calls for heat the ignition sequence starts and the blower motor will come on. First of all the "sail switch" which is simply a microswitch with a little paddle on it that sticks out into the airstream of the blower should close and tell the circuit board that the fan is running. Then the ignitors start firing to get a flame going. Once the flame is started the thermocouple detects the presents of flame and tells the circuit board to keep the propane valve open so that the furnace keeps running. If either of the sail switch or the thermocouple are bad the circuit board will go into shut down and eventually lock out modes. The fan will continue to run until the "gas" is purged from the combustion chamber for a set period of time.
Here is the really great thing about older furnaces (like most of them from late 90's and earlier) The circuit boards will continue to try and light the furnace for ever more to the point of leaving the blower motor running pulling 7amps from your batteries until they are dead. Not only is the bad for the batteries but also the blower motor on the furnace as they don't last as long as they should when they are running on low voltage. But I digress.....
My first suggestion would be to get a new thermocouple and try it again. If this doesn't solve the problem I would probably go with a new circuit board and or try and test the gas solenoid/valve to see that it is working corectly. Both items are easy to change out but I would only replace the circuit board with one from www.dinosaurelectronics.com
. Their boards are far superior to the OEM products and will even fix the previously mentioned problem of failed light up by locking out the furnace after 3 failed relights therefore saving your batteries, much like the new furnaces do.