I'm going to go out on a limb here and risk exposing my ignorance.
A VFD is designed to power a motor. There are many 3 phase applications that don't involve a motor, where a VFD will not work.
You can use the VFD to drive the motors on the sinker, but you cannot use it to supply power to the control system. VFD's use a bunch of "tricks" to do their magic. Motors don't mind the tricks, but many other systems will have a problem with it. In other words, a VFD should not be used as a general 3 phase power supply. It's possible that if you left the VFD locked at 60hz, it might work, but it's still converting AC to DC and then using something similar to pulse width modulation to create the 3 phase supply. Most VFD's are "smart" these days, and it will probably figure out that it's not connected to a motor and might just throw alarms.
If you need a 3 phase power supply for the entire machine, a rotary phase converter is the way to go.
That's exactly why we used a (CNC qualified) rotary phase converter for our CNC back in Texas. Our simple rotary phase converters were not good enough, but they worked great for the manual machines. The one for the CNC was a 25HP with a ton of support electronics to clean things up a bit.
Michael