The #2X56TPI button head screws usually take a #6 TORX bit. If you break a tap in a Titanium handle, you can dissolve it with battery acid. I like to take a knife far enough apart that I can drill and tap a through hole. In plastic handles I add Helicoils to reinforce the machine screw thread interface. If you cannot drill a through hole, drill a deep one so that chips have a place to go, and you do not bottom the tap.
The percent thread is determined by the ratio of thread diameter to thread length. The shorter the length of screw engagement, the higher you want the threads in the hole. The Machinery Handbook has the charts that let you pick the optimum tap drill size.