For this experiment and project, I would remove both wires on the button and have that button work only with Arduino - no duino, no functionality. So whatever connection it may have with the car, it would be removed.
I already have a fuse block with 6 positive and 6 negative terminals for whatever tinkering electronics I plan to put in the car - can I use this as my ground? Arduino will be powered by a standalone DC converter, 5Vdc.