I'm suggesting that my guess is that all the button does is close the circuit when pressed- to the module it is just a HIGH.
I'm guessing the traces coming into the button are essentially just Vcc and Gnd. If you can figure out which of the traces is which and solder some wires on, I thinking maybe Arduino can write HIGH to mimic a button and fool the module. Or maybe you take the wires into a signal level transistor and have Arduino control that- it's an electronic switch.