ANOTHER UPDATE
Did a lot of digging under the dash, only to find out the bypass module was tucked up under the steering column, just out of sight on the passenger (right hand) side. Once I found it, problem was obvious -- connector was half out.
Plugged the connector back in, started with the key. Everything is good.
Also, I cleaned the multifunction switch again, this time with lots of contact cleaner. Relay still clicks occasionally on its own, and often a quick double click when releasing the brake pedal, but a HUGE improvement over what it was.
At the very least, I bought some time before having to buy a new switch.