We determined Matt has a Tremec 3550, not a 3650.
The switch he took pictures of is the neutral safety switch located near the end of the tail-shaft housing.
We may just leave it disconnected since it isn't harming anything with the broken wire.
His reverse light switch was actually located on the left side of the transmission housing, and upon removal and testing it was definitely bad.
I jumped the plug, and other than one bad taillight bulb, the wiring was fine and we replaced the bulb.
We ordered a new back-up light switch from CJ Pony Parts and it should be here/installed within the week.
Thanks for the help!