Make a Center Post to use the Tail Stock.
Make the diameter a little smaller but long enough to use the tail stock.
Light pressure on the tail stock
DOCs should be very light .02.
Use a VNMG430 (330) insert.
Feeds and Speeds Very low.
I have done plenty of little parts like this.
If parts like these come in your shop it would be wise for your employer to buy a Swiss.