This isn't a specific answer to your question, but there may not be a separate wire that only turns the reverse lights on! BMW puts digital logic in the light panel, and sends a digital, serial signal to it, and the light module then turns on the light. This allows you to not have to run as many high-current (and thus heavier gauge) wires. It also means that the switches can be very light duty since they are not switching any current, only sending a signal to a computer, digitally. So, in this case, there's probably one 12vdc line and one ground, then a pair of small wires that carry a digital signal. The reason you need to code the i3 to be able to turn on the fog light is that the computer was not programmed to recognize the message sent from the switch panel. On my 2011 GT, the programming was there, but there was no switch to generate it! So, just swapping the switch made it work. IOW, the wiring is there (power and serial data), but the car needed to be programmed to recognize the message to do what you wanted. Why the USA doesn't rate fog lights, even when they're already in the panels in the rear of most of their cars, really irks me. Since I don't drive my i3 longer distances, I've not invested in trying to make the fog lights work as opposed to my GT, which is my long-distance tourer which is more likely to be out when the weather could turn dicey.
2014 i3 BEV, 2021 X5 45e
(The i3 will be sold soon, <17K-miles, interested?)