You have had excellent comments on this topic.
I ride horses on the way in for all my hunts, and when I start hunting, I take my J107 from the horse and strap it on, and start the backpack hunting part of the trip. What I do is all the load on a horse and just me and the rifle on another horse.
Riding with the rifle is not that comfortable on long rides specially, and worst of with a heavy rifle. The scabbard previously recommended looks very good, just put some weight on the other side of the saddle and you should do just fine.
Please keep us updated on your results and experience, maybe we all can learn something from this.