I think as a general rule static phase converters are not recommended for CNC machines. However, it sounds like all you are looking to do is power the spindle motor with the phase converter. I would think a static phase converter would be just fine for that.
The variable speed spindle is not a VFD. VFD is a variable frequency drive.
I'm farrrrrrrr from an expert on this stuff so wait for the "smart people" to confirm all of this.