The other thing for umpires to consider is if (as you said) the wrap is being cut off to cut the weight down on the bat. If so and it does lighten the bat, then we have an issue with the bat no longer being legal for use since it is less than a -3. If its a -4 or a -3.5, Im not sure how we would actually know, but its clearly tampering.
The only time I ever caught an illegal bat, was when a coach called his player back to the dugout and said "use this bat its lighter"........I asked to see the bat, it was not one we had inspected prior to the game.....it was a -5 .........I could not believe he said that out loud.. I disqualified the bat......better DQ'd before the at bat, than an out at the plate for using one...