Didn't you post a super group not long ago?  My best grouping rifle has been worked on extensively.  The rear barrel band has been fitted tight to the wood, so there's no tipping of the band from the pull of the sling. This necessitated a slight thinning of the swivel tab.  The band has been filed inside so that it just touches the barrel keeping it against the wood that hasn't been removed below it.  The barrel has been free floated, including the front of the receiver ring, except where the bands are, to about 1/2" behind the upper band.  The upper band has been worked on inside so that the two bands that wrap over the barrel touch on different sides, but otherwise one can see light under them.  Not wanting to see someone mess up a good thing, and since you said sling tension seems to be cause, I'd start at the lower band, getting it tight to the wood, without being overly tight to the barrel.  Then check it out again before proceeding with other modifications.