As far as traveling in the cold and keeping heat in the underbelly...? That depends on how long you will be traveling and what the ambient temperatures are.
If you travel 3 to 5 hours in 30 degree weather in the daylight hours, you will most likely have no concerns about the under belly, or the water freezing in any of your holding tanks even if you have no heat source in the underbelly.
If you travel that same time and distance and the temps are 20 below zero, you'll have problems with freezing water for sure.
Winterizing is not a difficult job if you do it once or twice a season. But to winterize every few days to travel with no heat and the unwinterize a few hours later when you reach your next destination can be tiresome for sure.
But, that is my suggestion for safe travel if you have any doubts.
My second suggestion is NOT to winterize with RV antifreeze in this situation, but to do a simple air compressor blow. If the ambient temps are not dangerous zone (like 10 degrees or lower), blowing your water lines is a good way to prevent damage and does not taint your water system with the RV antifreeze and then having to flush it all out.
Attach an air compressor and open each faucet until it spits air. This takes no time at all. No need to drain the tanks. For a 3 to 5 hour drive, you should have enough water in your tanks they won't freeze up in that time. And even if they do, nothing will get hurt because there is good room for ice expansion. Once you power up the camper, the furnace heat under the floor will worm things up again at your next destination.
Again, if the temps are (my guess) ... above 20 degrees, blowing the lines and leaving the rest will be fine for a 3 to 5 hour drive. Your water heater will still have some warmth, so no need to drain it either.
That my suggestion. Yes, it takes a few moments to blow, but that's much better than spending hours repairing broken water lines from freezing up.
Edit: Before "blowing" put your water heater in by-pass and make sure the water heater is turned off ... gas and electric. When turning the water back "on" (either garden hose or water pump), fill the water lines first (purge the air) ... THEN take the water heater out of bypass. Purge the water heater (it will have a small amount of air), and then turn the water heater back on.