We have done I15 many times in the winter. Dick's rv park in Great Falls is open year round (we've never stayed there).
We leave I15 at Boulder and take highway 69 & 41 to Dillon. These are good 2 lane roads, used by lots of transport trucks. By taking this route, we avoid the 2 passes by Butte.
There is a campground in Boulder, MT - take the exit into town & it is just past the service station. It iis behind the carwash. I think it is called RC's.
We try to get to Dillon the first night & we stay at the Comfort Inn. They have electric available in the parking lot & allow RVers to stay overnight for $10. There is a washroom in the lobby that you can use.
Near Downie,Idaho we have stayed at Flags West truck stop, exit 31. They have a small campground & if you buy $10 worth of fuel, your stay (with electric plug in) is free.
South of Salt Lake City at Nephi, there is High Country rv park. Take the exit for Flying J and the park is about 1 1/2 km north on that street. Good access & washrooms.
Watch the weather and try to leave at least 24 hours after a storm has ended & when you have a 3 day window of clear weather.
We carry chains which are only to be used to get off the highway to a safe place. Luckily we have never needed them.
Drive safe & have a wonderful time.
Merry Christmas
Once you get to Mesquite NV you are out of any risk of icy roads and you can relax & enjoy the desert.