Are you saying to put the sling on the horse or still on your person?Personally, I’d recommend a sling for your bow and forego the idea of strapping your bow to a backpack and wearing it horseback.
Bow scabbard or sling on bow with arrows and quiver removed. One year I took header off horse when it spooked. As falling i was able to get bow off shoulder and hold up in air not landing on it. It was Rodeo