To the OP:
On your car, you also need to be sure BOTH fans work. And on the BRM (and all A5 cars, no matter the engine, for that matter), the fans require some refrigerant pressure to kick on, even low speed. So if there is NO pressure being created, no matter the reason, the fans will not come on. The system can have a full charge, but the RCV in the compressor is so bad or so weak that it just cannot let the compressor make enough pressure to make the fans work, and of course the system won't remove heat like it should.
With a suitable scan tool, you can go in and watch the A/C command and pressure value, although I do not often even spend the time with that anymore. I keep a stack of RCVs on the shelf. Most Sanden compressors (which is most of these cars) use the same clip-in RCV. They are pretty easy to change. I evacuate the system, look at the amount I took out, if it shows close (2/3 or more) of a full charge, I swap the RCV and recharge the system, confirm the fans kick on, and that the system cools. I also verify the cabin filter condition if it is not a car I regularly see.
Fairly simple process. It is just into July now, and I have already replaced probably 20 RCVs on various VAG products so far this year.