Energy comes from the line inputs. The ampdraw comes from the capacitance charged by the line inputs, the reduction of startup peak is only due to the time reduction of the compressor ramp up, and the back emf charge of the motor in rotation.Yes the hard start kit increases the starting amps but I would argue that energy comes from the capacitor. I installed one in a 2.5ton home unit and dropped the start up peak from the line by around 25%.
I am not contradicting your claims, merely pointing out the source of the results.
Hard start capacitor kits can reduce loads on an inverter, but not always, and not for the reasons it would seem.