Hi,
I know someone who does run one A/C through his 24 volt Trace 4,000 watt pure sine wave inverter via his 100 amp 24 volt alternator. The bus also has a 200 amp 12 volt alternator, and he does not overload either. However it does put stress on a 12 volt alternator and it's belt to run it at maximum amperage for hours on end.
I would not suggest it, however with the right alternator and upgraded belts it is possible. You must be using the pure sine wave inverter, because the compressor is a high torque motor, and can not run on modified sine wave power.
Fred.