Eric,
Suggest you first use a pin or other sharp probe to pierce wire insulation at reverse light terminals to see if you have 12V there. It's possible that corrosion has gotten into the terminal lugs, resulting in high resistance. Also check the connector in the rear (check voltage on wires same way) where the line from the electrical bay terminal branches out to feed the 2 backup lights. If one light is OK, the problem has to be between that connector and the bad light socket since there should be only a single wire coming from a terminal in the electrical bay to the rear of the coach.
Steve