My apologies for reviving such an old post but I am in quite literally the identical scenario (one bulb lights, the other doesn't, I've replaced the down step transformer and the socket doesn't seem that corroded to me but I used contact cleaner anyway).
I even ran two copper wires from the socket to the bulb with power live to be certain it's not a contact problem. Despite my voltmeter showing 12V across the socket I cannot get any of my three bulbs to light, despite them lighting just fine with a 9V battery.
