The first thing I would do is attempt to get the Tekonsha Prodigy RF Wireless Trailer Brake Controller working on your 2013 Porsche Cayenne. I would make sure you have power on your 7-way using a circuit tester like the # PTW2993. Since it works on your friends F-350 you at least know that it is functional despite the issue with the lights.
A lack of power is what would make the most sense since the remote pairing has nothing to do with the specific vehicle. If you do not have power then... view full answer...