You have a Class A DP. It should be fine to run the inverter while driving. It shouldn't run down the batteries. That's what the alternator is for. The batteries should have been fully charged when you got to the CG.
On the 3 DP's I've owned I always left the inverter on all the time - even when plugged into shore power. I suspect the Allegro is the same. They all had a pass thru relay so that if the pedestal went away, the inverter picked up the slack.
If you are able to recreate the problem, check the 12 volts to the furnace and confirm propane is on and doesn't have to be bled.