Is there a way to tell my horse to get lost?
Solution 1:
The best way to preserve your horse while attempting to complete the "Spurred to Victory" achievement is to hitch your destined horse to a post, and then only use other horses. This is easily done if you are able to use purchased horse deeds.
To the point of dismissing a horse while actively using it, the answer is no. Your options are limited to leaving the horse a good distance away from the action, and then praying a cougar is busy napping or that the horse doesn't see anything shiny at the bottom of a cliff.