1. Leaving it out altogether
2. A "brush horse" button that you click and that's the end of it
3. Making the player do it with the mouse
In American Truck Simulator you don't have to clean your truck, but you do have to get gas. Since a horse's cleanliness is more vital, maybe a cleaning mechanic (or at least a button/corresponding animation) should be in the game. Then you have to address the timing, which environmental/time factors, etc. would cause the horse to get dirty, if taking the horse into a lake will rinse it off (wait can our horses swim? Need to check with the dev team). I think I'd have a hard time designing a simulator that would actually be fun. So many trade-offs to consider.