I think Lando might have the best solution. If you use heavier thread, the problem should disappear. It is possible you have a burr or something on the tool that is the culpurit, so you might want to try smoothing the edges down with a file or emery cloth. The Brassie is sized as well, so it might be the case that you are using the smallest size for large hooks, and that is causing the problem, but again a heavy thread should solve the problem.
If you really must pack using light thread, then you might want to resort to just using your thumbnail or a ball point pen instead of the Brassie. Luck.
I have had issues with brassie breaking flat waxed nylon. I always used a thread like GSP or Gudebrod GX-2 for my Bass flies. Kevlar is too bulky for my taste. For muddlers and Goddard caddis I used flat waxed nylon and used my fingers to pack, for Bass flies that I want tight, and float for awhile, the GSP works the best for me. Try to put a thread base down before you spin or stack, it might slip a little easier. Good Tying!!! Steve