Fact is ..... You don't need any bushings at all. . . Here's an outline of the procedure. . Possibly leaving out some obvious steps.
Put a dimple in the center of each end of the blank. . Small, shallow drill hole at each dimple.
Dead center in the head-stock .... Live center in the tail-stock .... Your blank in between .
Pressure from the tail-stock will keep your blank spinning while you turn to round.
When the blank is round and about 0.7 inch in diameter, you put it in the headstock and drill it to receive the brass tube.
When the brass tube is in, you mount the blank again between centers and turn (and shape) down to the correct size.
Your dead and live centers now make contact with the brass tube ends (assuming you have trimmed the blank to length).
Here's when you need to keep checking the size of the blank with your calipers .... at several points along its length, paying special attention to the ends of the blank where it will meet the hardware (nib and finial). . At this stage you'll be taking off VERY light cuts and measuring often.
If you have any questions, please ask. . Happy to tell you all I know and many others will help also.
And ... Big warm welcome to IAP !!!! . You have come to the right place !!!