Electronic – Run electric motor CW & CCW with DP-DT switch

I am quite confident that you should connect nothing to the brown, black and yellow wire group and 220-240 volts to the black and white wires with a 1.2uF capacitor between the black and red wires. The motor appears to be a permanent split capacitor motor (PSC). A PSC motor must have a capacitor connected at all times in order to develop starting torque and run in the correct direction as well as developing adequate running torque.

The brown-black-yellow wires appear to be connected to an internal temperature detection device that is intended to protect the motor from overheating. There appears to be an additional internal protection device that protects the motor from excess motor current.

Loss of the protection by the direct temperature detecting device increases the risk of motor failure, but you can not use it without knowing the proper voltage to connect to VCC and providing an appropriate detecting device to monitor Vout. The proper voltage for VCC is very likely a small DC voltage, not 220 V. The temperature detecting device is probably a thermistor or a PTC thermistor. You could search for motor protection circuits using thermistors and see what the usual VCC an resistance values are. I suspect the reason for this extra protection is to assure that the motor does not overheat due to high external air temperature, blocked air flow over the motor or accumulated dirt on the motor. You can reduce the risk by keeping the unit clean and free of air obstruction. Manually shut the fan down if the air conditioner fails to cool the air.

The diagram in the question shows a that the 220 volt AC power is supplied through a traic connected in series with the motor. The triac can be used to start and stop the motor or reduce the speed by reducing the applied voltage as explained in answers to this question: Speed control for PSC induction motor. Protecting the motor with a winding temperature sensor is more important when the motor's speed can be reduced. Reducing the motor's speed can reduce the flow of cooling air flowing across or through the motor. That could cause the motor to overheat even if the motor current does not exceed its rated current.

The motor is a single-phase induction motor of the permanent-split capacitor type. It is called permanent-split-capacitor because it is a split-phase motor with the capacitor permanently connected rather than connected for starting and disconnected after the motor comes up to speed when started. The diagram on the label that shows the windings and capacitor indicates that to people familiar with single-phase induction motors.

The label also shows how the power is connected for clockwise (CW) and counter-clockwise (CCW) rotation. That is described in the comment by Humpawumpa. The neutral of the power source is connected to the black wire and the hot wire is connected to either the blue or red wire for either CCW or CW operation.

To run the motor using a battery, you need a power inverter that converts the battery DC voltage to 220-240 volt AC voltage. The inverter must be capable of supplying 1.4 amps at 220-240 volts continuously. It also needs to provide a higher current for a second or so while the motor is starting. That higher current may need to be as much as 10 amps. Operating with a battery is not safer than operating directly from 220-240 volts AC.

Reversing the motor without waiting for it to coast to a stop would be the most stressful operation for an inverter. It would also be stressful for the motor. The motor may or may not be designed for that. The washing machine timer limits how it is reversed and how often it is started. The motor is likely not designed for more stressful duty than allowed by the washer timer.