If the inverter cannot start and run the air conditioner on battery power, with grid disconnected, there is no reason to have the air conditioner running through inverter pass-through relay. It just puts unnecessary stress on inverter's pass-through relay and causes lower AC voltage at compressor making startup surge last longer in time.
A high air conditioner startup surge current through the inverter pass-through relay and internal inverter wiring can result in a voltage drop from grid AC input voltage to inverter's AC output. The inverter response time to follow a dip in grid voltage is relatively slow so inverter may pick up much of the air conditioner startup surge current.
Phoenix, Az summer temps are brutal causing very high compressor head pressure. Hard on compressor, hard on condenser coils. You want a unit that has large condenser size for its btu rating to get the most heat exchange possible to lower pressures on compressor output.
Also make sure you have a 5 minute off timer to ensure a power glitch doesn't try to restart compressor with a high refrigerant head pressure.
