How does a bad power factor affect batteries with inverters?
The diagram below shows a basic inverter power circuit with a battery source and induction motor load. An induction motor is an inductive load because it has a magnetic field that must be maintained for it to operate. Since the magnetic field changes polarity every time the current reverses, it must constant exchange stored energy with the source. That circulating energy result in a lagging power factor. It is not a "bad" power factor, it is the power factor that the motor requires for its operation. The only energy that is used as a result of the lagging power factor is the losses in the conductors carrying the current.

If the circulating energy would be circulated between the battery and the motor, that would involve the motor adding to the battery charge during part of each cycle of the AC waveform. That might be possible, but it would add to heating and the wear-out mechanism of the battery. To avoid that, the inverter should have a diode to prevent current from flowing back to the battery. A capacitor is added to exchange stored energy with the motor.

Energy is returned from the motor to the capacitor through the diodes connected inversely parallel to the transistors. The result is a large "AC" ripple current added to the DC current. The current to the inverter has a DC component supplied by the battery and an AC component that flows in the capacitor.

If the capacitor and input diode are omitted, the battery is likely to fail or have a shorter than normal life. If the capacitor is not large enough it may fail or have a shorter than normal life. In either case, the motor may have difficulty developing rated torque, particularly starting torque.