I have spent many thousands of hours hunting, scouting and packing in camps on horses and mules. I have never seen anyone with experience carry their rifle on their shoulder.
Should also consider instead of saddlebags, use a small backpack on the other side of your scabbard. This will help...
When new rifles don't shoot, don't invest time or money. Send it back.. repeat if necessary. To many people try to fix rifles, fail then send them back. At this point you may share in the responsibility to fix.
I would make arrangements with the outfitter to pick up quarters the kill site, even if its a couple hundred bucks. FYI can't leave elk whole overnight. ( need to quarter ) I know this was not the question.