With A/C on fan speed one is normally activated by the IHKA.
Only Via BMW tester equipment can you check the operation of fan speed 2 without running the car hot.
Remove and check the water pump
These water pumps use a plastic impeller and decay over time, the impeller becomes separate to the main pump shaft, causing over heating at nearly all situations.
They are very stubborn to remove, there are 2 threaded holes on the side of the pump, screw in two 10mm bolts evenly and SLOWLY and this should push the pump away from the block.
If the pump breaks upon removal, it needs replacing anyway and goodluck, the pump bonds to block like crazy and requires several sharp blows to crack the pump into pieces.