I have 3 mill cutter heads. I find that the one for the 7mm did the same thing to me. I ordered the cheap apprentis (the smaller sized head) from CSUSA and use to just get that last little bit of brass. I only have that trouble with the 7mm tubes. I use inserts for all other kits so I don't have to switch pilot shafts.
On some of my 3/4" mills the ID of the cutting teeth is slightly larger then the 7mm tube. If I use it on them the tube goes up inside the ID a fraction and will appear proud. Using my 1/2" mill does not do the same. The cutting teeth ID is flush with the pilot bit and trims the brass flush. You should only be cutting enough to just shine the brass and not trim it but that's another issue.
It happens - the hole at the center of the blades in the mill cutter is very slightly larger than the diameter of the tube, with the result that the cutter leaves a very thin rim on the tube. Easily removed with a pen knife.
Given that my goal is to not cut away the brass tube, my solution is to simply be more gentle when using this tool. If I do happen to get into the brass tube, I mentally slap myself for being careless and quickly clean up the tube.