Is it possible to leave this oversize on the external diameter.
Follow your plans for creating the stepped hole, drill and tap the threaded section that you need, fit it to the shaft you need to "adapt", and then true up the old shaft on your lathe. Ignore the runout, if any, of your new part.
Now turn the OD and you will end up with a perfectly true adapter. If you want the adapter to be removable and reinstall to be true again then you need a nice clean square shoulder for the part to pull up against. If its a "set and forget" then loctite, or pin it if necessary.
Drill and tap works best for me on stuff this size. But I would be going along the above course to keep it true.
Cheers Phil