folks around here might scream, but oh well
take a propane torch and heat the shank slowly until it is full and dark purple
you can wrap a wet rag around the working end to protect its temper as well.
after it reachs full deep purple let it cool slowly, by sticking it in a coffee can of kitty litter/floor dry.
after it cools you can turn it in your lathe without much trouble, and if you are careful it won't warp either,, even if it does a touch you will be turning it down and back to true anyway.
done it many times, and made extended bits for manufactures by annealing and welding on extensions as well.
i suppose it depends on how finicky you are about precision and all that, but from what i gather a drilled hole is not thought to be precision anyway and needs to be reamed to the precise size if needed.
it don't sound like you are after high precision anyway.
that is what i would do if it were me, your mileage may vary
bob g