I think camping in a blizzard is a lot different than overnighting at a ski lodge once in a while.
As long as the OP is planning on skiing during the day, then the temps won't be so low that the camper needs to be heated while they are away. The daytime temps will be above freezing. So, trying to warm the camper during the day is unnecessary (things won't freeze) and unneccessarily dangerous. Just winterize the camper and use bottled water as needed.
When the OP arrives back at the camper after a day of skiing, use the furnace to warm up. Make sure you have enough battery power or a generator to run the furnace fan for the night.
If the OP is uncomfortable running the furnace while you sleep, just run it right before bed then hunker down with lots of good bedding, then turn it on when you awake.
If the OP can hook up to electricity, use a space heater with a remote control - that way you can turn on, turn off, and adjust the heat level as needed without having to get out of bed.
If the toilet sits directly over the black tank, then you can use a pitcher of water to "flush" the toilet after use - that small amount of water going directly into your tank from the bowl won't damage anything if it freezes. And as it's only one night, the next day you can go home and add more water through the toilet and then drain the black tank.