I hunted from muleback one time so, take my feedback with that in mind. I looked at scabbards but the commercial ones were made for compound bows. That wouldn't work for my bow. I ended up using a Three Rivers Strung Recurve case strapped to the saddle.
It worked but, it was certainly a compromise. I can also tell you that one day, it got moved around just enough to end up in the mules peripheral vision. The natural instinct of any animal (human included) is to move away from the object that startles it. That means that the mule was spinning in a circle with me on it's back trying to control it, figure out what was happening and solve the problem.
Needless to say, it was a rather harrowing experience. A two-piece bow would have worked much better. If I am to ride mules again, I would probably be wearing a backpack and tie my bow to it (3-piece takedown) taken apart and assemble when I got off the mule.
Another big problem is that with mules or horses, you can be rather mobile and get on / off the horse several times throughout the day. Assembling and disassembling a 3-piece bow each morning and evening is one thing. Relocating to find elk and doing it several times throughout the day would make the cost of a 2-piece bow worth the investment. I wish more bowyers made 2-piece bows.
The leather scabbard shown (or something like it) could work well but, I'll bet it would cost a pretty penny.
It's not easy to predict exactly what may happen when on mule / horseback. You certainly need to be careful and the easier it is to disassemble the bow, the better off you'll be. I even broke a carbon arrow as the guide cinched things down tight so nothing was able to move and frighten the mule. Despite that, I still had one frightening dance on the back of that mule.
Hope this helps.
homebru