The wired chime is probably pinching all the juice. Like, you can't just split a cable and expect the same voltages to be sent through the 2 sets of wires.
Personally, I'd buy a double-acting button - so you can have the wired chime and wireless hack chime running seperately, but in parallel when you press the button - like having 2 seperate buttons, one for each chime. In fact, you could just get 2 buttons, stick them next to eachother and stick something on top, so both buttons are pressed at the same time.
To do it with just 1 single acting button, you'd need to split the voltage, or use a relay, or do some other crazy stuff... better to just get a double-acting button and keep the electronic side nice and simple.

Health, Ammo, and bacon and eggs!
