I wasn't quite sure of what the button did at first, so I thought it'd be best to keep the Arduino separate from the garage opener remote.
Turns out, all the button does is connect the circuit to ground ( battery — ). I'm actually curious as to why they place the switch between the radio electronics and ground, instead of between Vdd and the electronics. Doesn't matter since this is DC?
I might try to run everything off the 12v battery, with a voltage regulator for the Arduino. That way I'd get common ground and could use a MOSFET as a switch?
Thanks for the tips!