May I suggest using a device called a "Roller Box" (when used on a Capstan or Turret lathe)?
Model Engineers tend to call the device a "Running Down Tool" because it is often mounted on the Toolpost of a Centre lathe, rather than on the Turret of a Capstan or Turret lathe. Mounting one on the Tailstock of a Centre lathe will keep it central (which has to be done manually if Toolpost mounted) but may reduce the length of material that can be machined if the Tailstock does not have a through bore, and the Saddle reduces the proximity to the chuck.
A Roller Box carries an adjustable cutting tool, and two supporting rollers, but can be of two types.
It is effectively a travelling steady with a cutting tool mounted on it
One has the rollers leading the cutting tool (riding on the raw material), or the other type has the rollers trailing the cutting tool, (riding on the freshly machined material). The latter type is the only practicable one if the raw material is not truly circular, or is off centre.
The trailing roller type enables the machined diameter to be concentric with an existing smaller diameter.
The leading roller type enables the machined diameter to be concentric with an existing larger diameter.
Since the material is supported close to the cutting tool, it is invaluable for reducing the diameter of slender material, especially over a long length, with minimal taper, or risk of bending.
MUST get round to making one, (or maybe both types) one day! Add to the list of "Tomorrow" jobs!
Howard