Brody mate you sure went to a deal of bother there the chuck you are using could have a small set of expanding jaws for a nominal price that would fill the bill more accurately for years I note from the photographs that your wooden one is a bit hit and miss at keeping right angles I can guess on a ring it may not be critical. As one of your greatest admirers I did hesitate to speak up there also is a captive pin device you can make for this purpose much along the lines of the devices for rounded end pen blanks that cope with very small inside diameters easily made in steel and highly repeatable in use.
Kind regards Peter.