Does This Sound Crazy... Water Softener

I think you would do better with filtration than a conventional resin tank. There is going to be a great deal of engineering and difficulty trying to orient a brine tank in the basement. Another issue might be that even if your coach was able to handle the electrical demand of the timer and valve motor, you would need to have a fairly good flow of water available at the softener. All of that would mean you need to be hooked up on a city supply. Depending on the quality of the city supply, you may need no treatment for that. The real killer of this would be having any water supply devices in an unheated bay while parked in below freezing weather.