I lived in Lake City for a few years and rode Snow Mesa (top of Spring Creek Pass on the north side) in February once. That section of the CT isn't on the bike CT route since it dead ends into wilderness area, but it does have a groomed snowmobile trail up it in winter.
Without a fat bike, I had a lot of hike a bike, even in the best trail conditions... but still worth it just for the downhill and the pictures...
Don't hang out too long here.
The nice groomed corduroy was usually rideable above treeline on flat ground. Slight uphills were sure hike a bike.
Then across the street on Spring Creek Pass going southbound, there's a network of groomed snowmobile trails for a ways, some go to Slumgullion Pass and partway down the hill (Rambouillet Park).
There is a yurt system up in there (4 of them I think) that would be fun to link up in winter.
I've also done a bit of snow biking up the Rambouillet Park snowmobile trail too.
My suggestion is head for the top of Spring Creek Pass with a fatbike, backcountry skies, and camping gear (or some yurt reservations) and see what the snow conditions are like. Then in the likely event that the snow biking isn't great, just go skiing. Baldy Cinco (above Snow Mesa) is a good ski peak, and there's lots of good tree skiing near the top of Slumgullion Pass.