Tamid, Great ride. Do what you want, but I've been riding long enough to know that sooner or later you are going to break that nice shotgun. It needs a hard case !
I think the gun across the handle bars is a recipe for trouble - to the gun, or to your body parts when it hangs on something. If this was something I was going to do lot, I would invest in a saddle scabbard that lets it ride with butt pointed rearward and up. Might take a few trial and error adaptations to get it right, but someone must have done this already.