This might work. The collars might make finding center easy.
Make your cut where you want it.
Put the collars on temporarily, the inner fixture should be easy to center inside them.
Turn the inner fixture upside down on the collar, a pointed end on the center threaded rod can be used to mark the center of the hole to be drilled.
The worst case would be to drill one or both halves of the shank with slightly oversize holes and use epoxy to fill the gaps between the wood and fittings. With the parts assembled, clamp the shank together aligned like you want it until the epoxy cures. Use wax or grease on the inner threads and between the two halves so you don't end up with a one piece shank again.
If it's possible to do without being unsightly a thin leather or rubber washer will allow the joint to be tightened snugly by hand without over tightening it.
It should be a fun project.