Hey so I experienced this problem and found the issue searching other forums. I'll see if I can find those links, but bear with me.

BMW AC compressors of this era have a mechanical valve inside that can fail, and make diagnosing the problem impossible. The compressor clutch will turn on, but the AC will blow warm until you rev the engine. Because the system thinks the AC is working, you'll see a bunch of electrical anomalies since the pressures aren't what the system expects.

So if your problem is that the clutch is engaging, but you don't have cold air until/unless you rev the engine OR it takes a long time to cool off, this valve inside the compressor is bad and the compressor needs to be replaced.

Diagnosis of this problem is almost impossible since it's a physical valve inside the compressor so no error of this problem will show up in the scanner.

I'll see if I can find a link to the thread that explains this better. But long story short you need a new compressor.