Wiring for a westinghouse electrical motor?
With the given information and discussion, I think the motor will operate with the following connection. The capacitor may not be needed. It the current in the start coil can be measured during start and it is relatively low with a capacitance value used for similar Hp motors, it might make sense to try a direct connection from the switch to the power line. There would be a risk of burning out the winding fairly quickly if it is designed for use with a capacitor.

Various electrical diagrams for washing machine are available on the internet. Some show motors that use a start winding with no capacitor and some show a capacitor. The extra centrifugal switch is shown on some as switching a function external to the motor. The extra contact on the starting centrifugal switch is shown as feeding power to an external function when the motor is up to speed.