unless its Illegal to NOT use a case, I would never use a case , scabboard or the likes, and my reason is, WAY too many vibrations from atv, to the gun and any scope on it, or even just sights
having had a gun shop, and seeing damages first hand and stories of point of aim going off from being in case's and such, or just stocks getting all marked up(not that I'm a pretty gun type)
I personally on all my ATV trips, I use a sling for rifles and shotguns(AR's too)
if I LEGALLY HAD TO, have a gun in a case, it would be a full size HD hard sided gun case, I most likely have it mounted on my front rack, and have a thick piece of closed cell foam under it, between case and rack, to do as much ANTI Vib as possible to things, I'm sure it would stick out past wheels on each side, but being right in front of me, I could see it and steer around things as needed
I just ain't a fan of having a gun mounted in any gun holding device I have yet to see, just too many vibrations for my likings
a double sling set up would hold it on my back and be able to move freely
NOW< another idea, might be a back pack with a gun holder
that would for sure keep gun mounted solid and on your back and still not get beat with vibrations??
in a pinch, I gather you could strap the pack to the rack and have the gun on top like, and the pack could save it from a lot of vibrations too?, but make sure it mounted very well to that pack or????