Can't read German but it appears from the pictures removal of the radiator is called for.
I've done this a couple of times. The first time I fished it from the top between the radiator and the engine. It was tough. I then found the easier way to remove the power steering pump pulley first. After the pulley is removed the alternator drops out to the bottom. In either case removal of the radiator is not needed.
Don't mean to make is less of a job. Getting to the battery cable connection to the alternator is a PITA. BMW made a major improvement on the M62 engine by moving the oil filter housing to the car frame instead of attaching it to the engine. Then again the alternator became water cooled. I hope I never have to replace it.
'01 540it, 6/01
'03 325i 5 speed, 9/02
'10 535ix. 9/09
'10 mini 6 speed
'15 mini countryman 6 speed