Not 'any' class C can sit on a freezing mountain top and not freeze the tanks.
However many can. My Bounder can sit in 19F weather without any damage. Many class C and class A have 'enclosed' tanks, that are in the basement, with a 1" heater duct into the basement, and that 'should' stay above 32F.
I have a 415 watt solar panels and 4 golf cart batteries.
For winter heat, I have a Olymmpic Catalytic heater, 6,000 Btu's. I put in a tee and quick disconnect for the heater, and store it covered in the closet. You might consider the smaller heater in your tiny RV. Or leave the windows open further. I have MAxx Air covers on all my roof vents, so I can leave them open in the rain, or driving too.
If you stay plugged in during storage, it will keep the battery full. Just check the water levels every 2 months.
Check your tries for clearance. I used cables on my Ford Edge, and they worked fairly well. Chains will require more space between the tire and RV body. I would put cables on all 4 rear tires, because if you only do say the outer pair of tires, then those tires outside diameter is more than the inner tire without cables on it. You do not want the other tire scrubbing and traveling another revolution per mile.
If you have 'exposed tanks' under the class C, then you should consider camping with wine! Buy a gallon of the really cheap stuff, and put it in your fresh water tank, it will not freeze until it is really cold. Put about 1 quart in the black water tank, and some in the fresh water tank.
You will barely taste the wine in the fresh water tank. And it should keep it from freezing until the water is around 28F.
Good luck, Have fun!
Fred.