Easiest way to diagnose that its the compressor vs the fan itself is it will cool right off the bat with a bad fan then go hot and not cool again till you get up to highway speeds. Well as right off the bat as any bmw does. I haven't seen one yet that cools instantly on the first start of the day even when new. Takes about 30 seconds for it to get "happy" at cooling.

If its the compressor it wont cool at all until x amount of time elapses. As the compressor gets worse over time the delay will get longer and longer for it to start cooling. Sometimes reving the motor up gets it going. Reving the motor up will not get a bad aux fan going.