Forget the rewind IMHO, it will eat your cash.
Better, a VFD is probably "better" but it depends what your specs are.
Rotary can be basic and cheap...you just need another 230v 3phase (many motors are dual voltage 230/460 and will work just fine) AC motor, the same or larger than the lathe.
There is a little electrical work to go into a rotary to offset its lower cost. Also a rotary is more easily adapted to additional machines that need 3 phase power.
A VFD is plug and play, two hot lines coming in, ground the case, and hook up your motor, 3 wires and a ground. The VFD is more complex, having some programming parameters but the ones I have been involved with had a basic enough set as a default to run a motor at a steady 60Hz. Variable speed can be as simple as the control dial or arrows on the front of the VFD. Make sure you get a "200 volt class" VFD for your household current of 230vac.
This is a basic introduction, your best bet is to read old posts, ask questions on this one, and the help is all right here on the forum.
-Matt