How do you prevent a speaker from interfering with an array of buttons that use a voltage ladder?
If you had an oscilloscope, the first thing to do would be to look at the signal on ButtonsIn to see what's happening there. Since you didn't say, I'm assuming you don't.

It's possible that the speaker current introduces enough voltage offset into the shared ground wire to confuse the button circuit. One quick thing to try would be to connect a capacitor (0.1 - 1.0 uF) between ButtonsIn and GND on the main CB board. Since the interference is AC, this should eliminate it.