To VFD or Rotary Phase converter

I've never used the rotary phase converter setup, so I can't comment on that. I put a VFD on the mill I bought a few years ago and it's worked beautifully. They've become very inexpensive and just seems like the easiest way to go.

If I'm understanding right, you're considering paralleling both motors off one drive? Technically that's okay (think some sort of machine tool that runs two identical motors for some operation, where they always run together and at the same speed) but in your case you'll have to install a disconnect between the drive and each motor, and then remember to jockey them on and off depending on which tool you want to run. I guess if you accidentally leave one on and try to start the other tool the drive would detect the overload condition and fault out. Still it's kind of a hokey setup.
As cheap as they are now, why not just buy a second VFD for your lathe? I went with KB Electronics for my mill. They have a 240V single phase input at 2HP for $238

KBDF-27D - KB Electronics - AC Drives | Galco Industrial Electronics

I put mine in a small enclosure and added a pilot light so I'd know if I had the power on:
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drive box.PNG