[cart around]

As long as the solenoid is activating while in reverse, (I don't see why it shouldn't) The only thing that changes is the direction of the starter, and the FnR switches direction.

Do this test with the power off. Do Not step on the pedal.

Set the switch to reverse. Set your meter to ohms. Check for continuity between the big switch lugs A-D and B-C. That tests the switch that you already cleaned.

( The switch is looked at from the back. A on top, B on the passenger side, C on the bottom, D on the driver's side.)

If the solenoid is not activating while in reverse, there is a microswitch located on the lower passenger side of the FnR switch. That is the one not shown in the manual. Check that for continuity. You could also just check for power at the solenoid small terminals while stepping on the pedal.
That microswitch should make contact in both directions. It is connected in series with the key switch, and pedal switch then to the solenoid.