Fleece-lined is the cat's ass. A filthy, disease ridden, diarrhea inficted cat's ass. They would absolutely ruin a rifle here, and other places as well. I'd sling a rifle while in the saddle before I used one of those.
Here is an axe on SonnyD(to help balance the rifle) opposite the rifle. The rifle is a right-handed bolt so has to go on the off side. I've seen too many broken stocks from carrying butt forward in *this* country. It'll catch a tree and snap.
A rifle slung on Mac:
Axe on the Creamsicle:
Try to avoid affecting your leg, and the horse's motion. Make sure a bolt gun doesn't allow the bolt to poke the horse. That's why I'm obligated to have it off-side, while I'd prefer to have it on the side I mount/dismount. Also I prefer to not have the gun ride on the scope, or I could hang it on the near side.
These scabbards have leather straps and I just feed them through the girth rings.
A lot of what I posted really doesn't apply to your case as your are hanging a LA. But some thoughts to keep in mind.