If diagnosing isn't your strong point and you want to use the parts cannon then replacing the dash control (IHKR) might be a good guess especially if you can get one for cheap at a wrecker. If you get lucky you might even find a good compressor, maybe one that has been replaced.

I think the manual air might not use the variable displacement compressor so something else to check into as well.

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1988 325is (purchased new) sold
2004 X5 3.0 2005 X3 2.5
2008 X5 3.0 (new to me)