Leather is the trick. Get the thickest stuff you can find (within reason). I’m using 8/9oz (3.6mm) vegetable tanned full-grain cowhide leather. After clamping on your first barrel it’ll ‘form’ to the clamp, which makes it easy to place. I haven’t had any trouble since using this (albeit with a Viper vise), even clamping onto barrels where the fluting is fairly aggressive at the clamping spot. Just pick up a sheet of the leather and cut the strips slightly deeper than the clamp block’s barrel channel, and as long as you can without interfering with the clamp bolts.
I don’t even bother with rosin and I’ve yanked and installed a lot of barrels running 70-90ft/lb without a single scratch to any finish on any of the barrels. I’d been messing around with layers of masking tape before and beyond making a mess, sometimes the adhesive would slip and it’d cause slight marks on the barrel. This was primarily a problem with fluted barrels, but leather is just better for everything I’ve run across.