Here is an idea, drill the hole through with the drill press. Then make one of gadgets on post 13 of this thread. Using metal lathe for wood spindle (Celtic whistle) longer than lathe bed

The hole will be centered on the threaded gadget that I call spindle driver. You will need a tap so that you can tap a blank to screw on the headstock threads. This way the hole will be centered between the headstock and tailstock. I used this method to turn musical flutes and duck call bodies.