Disassemble and turn thee bottom part of wood off to the brass, carefully and re-glue a like piece on and turn.
Below is a pict of how I make a laminated/segmented section of pieces. For you - find a matching color of wood to the part being taken off, at least an inch or two long and drill to the size of the tube. Then cut it thicker than you need. After gluing it on, carefully cut the excess off and turn. Since it is a small/thin piece, I would sand it down to NEAR size rather than use a tool, then carefully use your chisel of choice and take tiny bites to get it to size, OR continue to use extra fine sand paper to bring it to size next to the body of the blank and re-finish.
30 Pieces of Silver blank construction layout pict
www.penturners.org
A note on this, new finish on areas that have been sanded usually show up as a hint of a different color than the older wood with finish already on it. For me, to solve that issue, I carefully turn (with a very sharp tool) the finish off. By turning with the tool at high speed and very light touches with a freshly sharpened tool, I can scrap the finish off and have a smooth blank. The PURPOSE of this is to keep from having sanding dust spread to light and dark areas of the blank. I, and many others have done method this with regular success. Sanding segments creates smearing of different colors of wood (and metals).