The resistance readings are all as expected. Yes, orange/black is the main winding and red/white is the auxiliary winding with the capacitor. This is correct for 120V (low voltage) rather than 240 (high). Therefore, the motor is reversed by exchanging the two connections of one winding relative to the other. In the original Furnas switch, operating the handle connects the capacitor winding to the supply via terminals 1 & 3 in the same orientation regardless of direction, but connects the main winding via terminals 2 & 4 to the supply in different orientation for forward and reverse.
Forward: Hot -> 1 & 2, Neutral -> 3 & 4
Reverse: Hot -> 1 & 4, Neutral -> 3 & 2
Before you make any modifications, please consider that this machinery can cause injury or death in the event of malfunction. You must take responsibility for compliance with all applicable local and national codes and law. For example, if you modify the controls and a malfunction occurs that makes it impossible to stop the motor, and a person is injured or killed as a result, you might be held liable for the injury as the designer and installer of the new controls, even if the malfunction was not due to a fault in your own work. As contributors to a public forum, we accept no responsibility for the accuracy or sufficiency of any information that you might use. Nothing I contribute to this thread constitutes instructions to carry out work; it is for information only.
My approach would be to get a ready-built, reversing starter for a single-phase, 6-wire capacitor-start induction-run motor of sufficient hp rating for your motor. It would come with two contactors interlocked (to prevent one operating before the other has released, which otherwise causes a short-circuit). Both mechanical and electrical interlocking are usually provided. Then, I would connect the output of the remote control receiver to the forward and reverse control inputs of the starter. I would disconnect the internal holding contacts so that the starter contactors only operate when there is an output from the receiver.
Useful explanations and schematics here:
Interlocking Methods for Reversing Control (Basic Control Circuits)
www.industrial-electronics.com