Running the A/C compressor directly from the ICE will always be more efficient than using the ICE to charge the battery and then converting the battery charge back into mechanical energy to run the compressor.

But then you must consider that the battery is sometimes being charged for "free".

When you cycle a compressor on and off to modulate, moderate, the liquification/flow of refrigerant it will take about 2.5 times the operating energy to restart it against the previously built up head pressure. In other words you would need a ~3000 watt inverter to run that 10,000 BTU A/C.

Which is undoubtedly another reason Toyota uses the variable speed electric A/C compressor drive. Variable speed equals no BUMP LOAD (re)start effort.