Many problems here. First off a standard single phase induction motor is not synchronous. It develops torque through slip just like a three phase motor. However, most single phase induction motors have switched start capacitors and possibly a run capacitor as well to start and maintain a rotating field. These need to be dealt with. I have never seen a VFD that was compatible with a single phase motor and I have literally installed about 10000 of them. In fact I have three operating in my home shop and four spares on my shelf.
Single phase sourcing VFDs is common practice, but attempting to drive single phase motors with most VFDs will cause a phase imbalance fault.
In addition if you are trying to control flow in a centrifugal pump with a vfd it will be difficult due to the flow characteristics of centrifugal pumps. If you are trying to save energy cost, a simple throttle valve at the outlet will work. The pump draws less power as it does less work on the fluid. Avoid throttling on the inlet because it can cause cavitation and pump inlet erosion .
I don’t see a VFD as a viable option in your situation.