OK, so the brass tube is already cast in ? . Presumably the tube was painted before logo was applied
In that case you just turn it as normally, polish carefully to remove all scratches & done !
I have done heaps like this with beautiful results.
Turn as carefully as you can to minimize surface scratches. . Sand as necessary ... I would use all grits 320 to 2000, or you could substitute MicroMesh, once you get above about 800 grit, and use all MM "grits" up to the finest you have.
For polishing, I use four polishes in succession. . Start with a coarse liquid automotive scratch remover (Meguiar's brand is what I use). . Then I use the Novus 3 followed by Novus 2 (polishes available from ExoticBlanks.com and other pen supply vendors). . Finally, I use another Meguiar's product, Plasti-X, to do the final polishing. . That's just my routine, and others might have equivalent processes. . I do all this by hand because I don't have buffing wheels.