I've been on four lengthy horseback trips in rugged mountain country, and there's no way a scabbard would've worked, much less a hard case. (Too much brush.)
A sling works OK, but my shoulder got sore after a few days. What I finally learned was to let half the weight bear on my shoulder and half of it be supported by my cupped hand on a long stabilizer. To make it easier to hold the stabilizer I duct taped a tennis ball over the end; this fit well in my hand (and made it easier to maneuver the bow around branches and other obstacles). It may have looked a little goofy, but it worked very well!