I have one of the little tiny C0 lathes, more of a toy really but handy for doing odd very small bits like screws and rivets.

As an example I made a new steel backplate for a 4 jaw chuck on it, 62mm in diameter by 15mm thick, other than the tapped hole everything else was done on this tiny lathe just to prove it could do it.
took quite a while and would have been minutes job on one of the bigger machines but it proved it can be done.

The C0 is about 1/3rd the size of what you have.