Well GM service info is acting gay now so I can't look at a diagram, but I don't see why you couldn't tap into the gray or the brown/white wire. You can test it with a multimeter just ground one lead and check for voltage with the other lead. Not really sure about the ground wire on that switch.
I wasn't thinking about it being the starter cicuit so I guess the 2 pole switch might be dead without the key in the start position.