You will need tire chains (required by law) on some of the mountain passes if it snows.
That alone should dissuade you from going in winter;
Living in the MH in the winter will not work well if you plan to have water unless you find a way to keep the water from freezing like a heated compartment. How do you plan to dump when it is freezing. FlyingJ may well be your only option for dumping. Unless you have a rig equipped for winter you will go through a lot of propane trying to keep warm.
Your first trip should be close to home for shakedown