I do the same but have found that if I use a soft wood I don't tap the threads, I just let the threads on the mandrel cut them

Use a home made mandrel consisting of a 3/8" bolt with a 1/4" thick "washer" threaded and turned tight on the bolt to where the threads stop. Head of the bolt is cut off, leaving about 1-1/2" un-threaded shaft. Chucked in the metal lathe, and "washer" turned to same diameter as the stopper hardware. This is chucked in a 3'8" collet in the wood lathe.

I use a brass threaded insert in the stopper, as I not only have better luck that way, but the parts are replaceable. Key to the brass insert is to put the blank in a vise on the drill press, centering as close as possible by eye. Then a forstner bit the same size as the stopper hardware diameter or slightly larger is used to face off the end of the blank. The blank is then drilled and the brass insert threaded in. After that, it's just simply screw the blank on the mandrel, bring up the tailstock, and turn away. Using the forstner bit insures the blank face and the brass bushing are in line with the mandrel and lathe center line.