You can run a VFD off single phase if you oversize it. In this case a 15 hp. VFD would run your 10 hp. motor.
You have to be careful that you don't buy a VFD that senses for all three phases present.

I've been reading that providing "tuned" 3~ power using a rotary
coverter will product slightly smother lathe results than making the 3~
motor work on 1~ power. It makes sense in theory, but I have no
experience to know if the theoretical is actually noticable in the
final product. For that matter, I've read that 3~ motors are slightly
better for lathes than 1~ for the same reason.

I would like to have the 5K RPM and so that will require a 1:2.86 belt
ratio rather than the 1:2 in your example. If I decide to go this
route, won't I really need a 3~ 15HP motor with full 3~ power to keep
in spec with the 10HP @ 3000 RPM example you gave?

I really like the idea of compromising (staggering) capacitors for the
self-start idle motor during the startup process. If anyone has some
suggestion details please let me know, otherwise I'll post pictures on
my "how to" after my painful trial and error period (probably this
spring or summer).