If using a threaded rod, nuts and washers does the driving washer need to fit into the steering tube or flush with it?
Ideal is if the washers are just a slip fit inside the ID, but the chances of that are about zilch and would take too much work to get them good enough.
Knock off a few thou' all the around the OD of the old race, and use it between the near race and the washers once the washers reach the end face of the bore.
Smaller ready rod will have more threads per inch, so is better in that respect.
But you will need big bore big OD washers to get the washer dimensions needed, as well as adequate washer rigidity.
Sometimes two thick washers stacked are needed re stiffness.
When there is a mismatch of ready rod OD to washer ID, just use some ready rod sized washers or next size up, to "adapt".
Hope this helps you.
By the way, once a race gets cocked in a bore, it too easily ends as bad news.
The race must be started square and kept square.
The ready rod method is better in this respect than drifting a race in.
The class of fit can make it easy, or a nightmare, depending upon what one has to work with.