In my case the coolant motor is a fraction of the Lathe motor , and the VFD started via the lathe drum switch.
I worked on the basis that adding a further load when the lathe motor was already running was not a lot different to starting a decent cut and I have the VFD set for torque boost to take that on board .
I dont know how it would handle two motors of equal size but certainly sounds like your application would be better handled by separate VFDs.
The fact that you may wish to use the extractor ICW another machine would also make me choose that route.