As far as turning there are inserts made specifically for copper , if you have many parts to make this would be the way to go.
When only a handful of parts are required I have had excellent performance with aluminum specific inserts turning large copper electrical conductors. I would not do so without flood coolant however.
If only making one part go slow and use sharp positive rake tooling, the finish doesn't appear to improve with speed in this material, for drilling and tapping lard oil works a charm but is nasty in many ways.
For holding I would pocket soft jaws 1/8' deep and face one side then turn as much of the OD as possible, flip then face to length and turn the rest of the OD as needed.
Like so, one face and most of the OD finished, then flipped and faced to length and the rest of the OD turned. 42 parts in 304 SS, the order was for 40 but I made 2 extra because they get a good deal of mill work done to them afterwards including some 4-40 tapped holes, if a tap is broken in one I would have to set up the lathe again to make 1 more so I planned ahead.