You need to check all of the state regulations about traveling with open valve propane tanks! I believe most states prohibit that!
Personally, I have never run with the furnace on. I expect the low pressure area on the side of the camper rig, could cause excessive draw, either blowing out the flame, or potentially damaging the side of the trailer from excessive heat. Worse yet, creating a propane leak which would cause a catastrophic explosion!
When we set up for an overnight stop in cold weather, I leave the slides closed and start the furnace. Helps even more if your camper has an electric fireplace! We then go out for supper. By the time we get back, the camper is toasty warm. Oh yeah, don't forget to turn the covers down on the bed , so they warm up as well! Also, don't de-winterize until you are at least in the Carolina's.
Bandaid