Re: Furnace while travelling?
Yes, run it. With a propane furnace heat system, that is your only real option to insure no freeze-up if you are running water in your system during winter travel. You will see some "no propane while driving" fanatics here for sure, but "most" people who winter camper with full systems use run the furnace while driving. Exceptions are on ferries, or in some tunnels where you'd need to close off the propane system for that stretch but they are always a short enough time to not freeze or ruin the food in your fridge.
I think you are doing the right math and checking on your weights and I think you'll end up being in OK shape. The 19.5's would be good insurance. If I read your post right, you'd be right at weight limits BEFORE the extra tongue weight of the trailer. Sounds like you'll end up about 750# over GVWR. If you have a WD hitch at least some of the weight will go to the front axle.
Seems like you'll be in pretty good shape......but the proof won't be until you load 'er up.