Your exit lamp is not the source of your bulb out indicator coming on. Those bulbs are not monitored.
The reason you have 12 volts on both sides of the non-functioning exit lamp is that it is not getting a ground through the door switch. It could be the switch itself or a broken wire going to it. The wires sometimes break from flexing when the door opens, however the rear doors don't get as much use as the fronts. If you have moisture getting in, there might be corrosion in the switch.
Try grounding the brown/gray wire to see if your exit light comes on.