Chris, you mentioned that the right rear light flashes rapidly, does the left side flash at the same rate, or is it slower ? Rapid flashing is usually a sign of high resistance in the circuit.
I don't know your mechanical aptitude, but if you take a volt meter and measure the voltage at the center contact of the bulb socket, it is the 'hot' wire, it has 12 volts when the light flashes.
The other wire should be brown, it is the ground connection, and contacts the metal base of the bulb. Check for continuity between the side contact of the bulb socket and a good ground somewhere on the frame, or the ground connection for the battery at the transmission.
Have you swapped bulbs to confirm that it isn't faulty ?
Is the tip of the bulb making contact with the center contact in the bulb holder ? I think you can pull it out a bit to make sure it hasn't been pushed in too far.