How to programatically simulate a click on a capacitance-sensitive button?
It depends on the internal circuit, but what might be possible is to short (for safety with some current limiting resistor like 300\$\Omega\$) the two terminals of the capacitive switch.

A short can be seen as an infinite capacitor, which would make the circuit think that someone pushed the button (increase in capacity).

If that doesn't work you can use an analog switch to switch an additional capacity in parallel, which would increase the capacity as well. Touch sensors have a very small capacity, so some picofarad should be enough to trigger the button.

That being said, fiddling with the circuit might break it quite fast (it again depends on the circuit and the algorithm used for detection), and make the button always pressed.