Whats the recommended test and inspection for a VFD drive motor which is coupled to a pump. How do you test for pump efficiency?

I would run the VFD and record data of the pump output VS speed of the motor. Make sure inlet pressure is consistent and model it against the pump curve. If its within 2% of the curve, i would say its operating correctly and efficiently.

For VFD motor it is also suggested to check its electrical current.
vibration analysis also helps in motor condition evaluation because it gives vibration amplitude at 2xelectrical line frequency but its magnitude limit really matters. Analysing softwares limit 2xFL frequency amplitude and we should follow these limit.
other frequencies include bearing faults, rotor bar/slot faults, etc also indicate motor condition.
for vibration analysis of VFD motor, it is recommended to operate it on fixed rpm during inspection atleast.