Capacitor start motors (I think officialy they should be called capacitor start/capacitor run) use a centrifugal switch to disconnect the start capacitor once upto speed, otherwise the motor will cook itself pretty quickly.

Having said that, a 3phase and VFD is probably a more practical option. If a high startup load is involved, you might need to oversize the VFD a bit to give a bit more current for startup purposes.