Building a VFD for a split phase (capacitor run) motor?
There are VFDs for single phase Permanent Split Capacitor (PSC) induction motors, and if you say there is a capacitor but no centrifugal switch, then it must be a PSC type. These VFDs for single phase motors however will cost you a lot more than $70, probably 3 to 4x more, so you had better plan on using your other motor option.