To provide an update for others who might encounter the same issue, I went to the dealer yesterday and the module refused programming. They had some theories -- could be wiring, could be that module 5F (the infotainment module) needs to be upgraded -- but they did not seem very confident they knew what the issue was.
I think the module is just seeing something different between my set up and the pre-wired Tiguan set up. I pulled up the Tiguan wiring diagram on Erwin and it presented endless rabbit holes I could pursue. Since my trailer lights work and I am not getting error codes in any other modules, I think I am just going to let it ride for now. Hopefully somebody smarter than me will solve this riddle and I can follow their lead.
I think the module is just seeing something different between my set up and the pre-wired Tiguan set up. I pulled up the Tiguan wiring diagram on Erwin and it presented endless rabbit holes I could pursue. Since my trailer lights work and I am not getting error codes in any other modules, I think I am just going to let it ride for now. Hopefully somebody smarter than me will solve this riddle and I can follow their lead.