I understand the situation.

The only rifle I had a barrel band swivel on, was my .458 2". That rifle's quite heavy, stubby 22" barrel didn't seem effected, but - I never shot it using the sling for support, only for carry.

When using a sling for support in shooting, I believe the sling should be attached to the forend and the barrel fully floated so as to not effect the barrel's direction nor harmonics. You must check to see that the forend does not bend over to touch the wood when the sling is tight.

It is surprising how little tension it takes to bend a barrel and thus throw shots.

I use a long enough loop on the sling so the muzzle, even of my 25" .275, sits at or below my head when I carry it. My right hand is on the sling (or stock if I am carrying it in front of me, to prevent the gun from moving or slapping me when walking.

The front carry is interesting, as the gun is then usually carried on the left side, with left hand on the forestock. If needed quickly, the hand is already where it needs to be to shoot the rifle. Simply place the right hand on the wrist as you rotate the rifle butt up onto your shoulder as the left hand aims the rifle and it is instantly aligned with your intended target s the butt comes into your shoulder.

I will switch to that method when in areas where game might be seen at any time - it allows a faster presentation than about any other carry, except for double handed. At least with only one hand, the left, on the forestock, your right is free for tree branch deflection.

True - the first 1 to 3 shots being accurate are the vital ones.
Best wishes