Mount Rushmore in the Winter

I would go as long as you don't have a schedule you must meet. Travel when the weather and roads are clear. Your Bounder should keep you warm, just ensure you have plenty of fuel for that heat (diesel or propane). Most important is that you will need to call ahead and find out if any campgrounds are open and even if they are, will they have any water available. I would suggest that your towed should be one you are comfortable with driving in the snow (4x4 my suggestion) or you could easily get stranded in your RV/RV park without much to do. You should also check with the various places you plan to visit. Is there anything to see at that time of year? Are the roads accessible? Are the walkways cleared? What are the average temps and snow fall during those periods?
Having given you all the reasons to go, I would not go during that time of year as I believe you would be highly limited by the weather as to what you could actually see and do.