I would avoid attempting to drive an P.M. AC servo motor from a VFD, I tried it with less than stellar results, I have not heard of anyone yet doing it successfully.
I would put a three phase motor on the lathe, unless there is already one on there, and feed it with a VFD.
If you want to use an AC servo or BLDC, you will need a suitable drive that will work in the velocity or tach mode, and possess the right commutation feedback signals that you require with a AC servo.
Some drives will accept an encoder to be read as a tach.
Al.
CNC, Mechatronics Integration and Custom Machine Design
“Logic will get you from A to B. Imagination will take you everywhere.”
Albert E.