After many past years of snow catting at Camp Hale, I would like to throw out a few suggestions for whatever they are worth:
-Staying in Leadville is my preference, although a bit of a drive to Camp Hale trailhead. Tennessee Pass road seems to always be easy and well maintained regardless of weather. Ski Cooper (small) is the only ski area in that area, so less traffic and crowds on that side. Lodging is more reasonable than Summit County side. A lot easier to maneuver trucks/trailers in Leadville at motels, grocery stores, gas stations and on side streets near restaurants/bars downtown. Cool historic, funky, old town, if you've never been there.
-Vail Pass trailhead is often a zoo, especially on weekends, with snowmobilers, so I avoid it.
-I would not advise trying to stay in Red Cliff if you are pulling a trailer. It looks like it is right on the highway, but isn't. Narrow road into canyon where town is. Suggest scouting it first without a trailer. Parking is tight.
-Camp Hale (north of Tennessee Pass) has an easy pull-off unloading area (seldom a requirement for backing-up a trailer there). Has always been plowed. Lot of room in that open valley to cat around or go up. We've often barbequed at trailhead.
-Nova Guides, located just north of there is a private snowmobile/dogsledding operation (in Pando) which may not welcome non-customers with trucks/trailers filling up their lot.
My two-cents worth of 'opinions' on the subject.