Matt,
Yes, it seems doable. Pretty simple three steps to accomplish it. Note: I have not done this.
While my presentation on trailer wiring
(https://aviation.derosaweb.net/presentations/#trailerwiring) doesn't touch specifically on a backup (reverse) light you might find the information within it useful. It has the information on all the various Euro and US trailer connectors and includes a schematic for a Cobra trailer.
As you noted there is an "Aux" pin blue wire in Cobra trailers that is listed as for a "fog light". My documentation indicates that this blue wire is not used inside Cobra trailers at least for your vintage.
Here are the three steps;
- At the front of the trailer connect the Cobra trailer's blue "Aux" wire to the US 7-pin connector's "reverse lights" center pin.
- At the aft of the trailer gain access to the Cobra internal wiring block located behind an aluminum panel (see the attached photos below). At the top is an unused blue connector which contains the needed blue Aux wire (+) and a white ground wire (-). So if you can find the matching Euro connector this project will be simpler. Let me know if you do find a source.
- Connect your new backup light to the blue Aux wire and ground. Make sure of proper polarity if using LED. It should work.
Let me know how it turns out.
Good luck, John (OHM)
Warning: I have seen several Cobra trailers with burnt wiring. Tread carefully. Test frequently with a voltmeter. Hopefully you read this far!