To VFD or Rotary Phase converter

Thanks for adding the extra info. Basically what the Variable Frequency Drives are designed to do is run a 3 phase motor of pre-determined size at either a fixed or variable speed. Because they use output transistors to create a synthetic 3 phase power, they can also provide a 3-phase output when powered by single phase input power. That's the upside to them, the downside is that those transistors don't like being energized with no load. In other words, they're NOT like a transformer, where you can have power applied to the primary (input) side and anywhere from full load to absolutely nothing applied to the secondary (output) side.

So with the additional components that need to be powered, a VFD probably isn't the easiest solution. I'm guessing the lathe's electronic controls will be coming off a transformer which will only need 2 legs of the 3-phase, so that you would wire to the power you're feeding the VFD with and not the output from it. The coolant pump is the tricky part. It will be cycling on and off and will provide an additional load to the drive. It was also designed to work best at 60Hz, which defeats one of the advantages of the VFD, being able to change the frequency and thereby the speed of the motor.

So the rotary phase convert is likely the best solution in this case. From what I know about them, they're more expensive but they provide the flexibility of powering all your equipment from one source.