Linda, I don't think you could run the heat strip during the day from the battery power. I think it would drain the battery--very quickly. Heat strips operate from major resistance. I've never run my furnace while driving. Don't know if it's recommended.
I'd leave the trailer winterized til I got far enough south to avoid hard freezing weather. Use a bit of water from a gallon jug to flush if you use the facilities on the way down, as you said "dry camp". No point in pushing it, and creating problems in your travels. We don't fill fresh water when we come home from North Carolina in November, till we're in Florida.
Have a great trip.
Sherry
Thanks, we thought that might deplete the battery. We will leave it winterized till we get to a warmer place.
Linda