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Here is another approach that you may find easier to accomplish.
Chuck a piece of brass stock and turn a taper to fit your tailstock. Part it off, leaving enough of the parallel bar stock to reverse it in your collet, and face it off.
After facing, “catch” the center with your sharpest graver, and make a tiny dimple to start your drill bit.
Place the taper in your tailstock, and chuck your #80 drill bit in a collet in the headstock.
Lock the push rod in the tailstock, and using a fairly high speed on the headstock, drill a hole about six times the diameter of the bit in the brass, by sliding the tailstock on the ways. A little spot of kerosene (paraffin for you Brits) won’t hurt any. Back the bit out and wipe it off often, to keep any swarf from clogging in the grooves.
WITHOUT REMOVING THE POST FROM THE TAILSTOCK< remove the entire tailstock assembly from the lathe and clean out the hole, and degrease it with naptha, followed by an alcohol rinse. Now. "glue" the shank of the bit into the hole you have drilled, using either crazy glue or blue Loctite. You now have a drill chuck that you KNOW is centered on the headstock.
Using the “sliding tailstock” method, drill the arbor for your pivot wire, and proceed to re-pivot as usual. BE CAREFUL ABOUT Rotational speed and drill bit feed! Too high a speed with too little feed will work harden the face of the bore hole, and too slow a speed and too heavy a feed will guarantee a seized and broken bit. Again, Kerosene is your friend, and be careful to back the bit out frequently. What you are doing is referred to in machinist circles as “deep drilling”. When you decide to remove the bit from the tailstock, a little heat will soften the glue sufficiently that the shank of the bit can be removed from your taper. That taper can be reused, but only by re-drilling it with a larger bit, because reinserting the #80 bit is almost certain to be off center. (I try to make a rule that when I use a blank a second or third time, the drill I insert in it is at least twice the diameter of the previous bit that I used, and I drill the new hole with the highest headstock speed and a very slow feed, to help it overcome any eccentricity of the pilot hole.