Is there a way to tell my horse to get lost?

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.