As the hole is tapped, the tap has to move forward into the hole. If the work is held in the lathe chuck, and the tap is held rigidly in a chuck in the tail stock, what's gonna move?
They make a nice tap-holder (see MSC) that's got a hole in the backside that slides on a 1/4-inch rod. Chuck the rod in the tail-stock chuck, rest an arm of the tap-holder on the tool rest, and urn on the lathe REAL SLOW. Dave Golber