[GuidoK]

Both advices are partly wrong imho.

For diff fluid, you need the 75w140 with friction modifiers (so 'limited slip' has to be mentioned on the bottle), as as far as I could find you have a clutch type limited slip differential. So that Red Line 57914 is a compatible oil. (there are of course plenty others. for example from castrol.)
The reason why clutch type limited slip differentials in general need 75w140 is that the clutch pack heats up the oil when engaged, so it needs a wider temperature range. (hence the 140)

For the transmission you need a MTF-LT-2 compatible oil. So the BMW 83222339219 is correct. But any oil with MTF-LT-2 will do (I think Castrol, Febi, Meyle and lots of others have one, don't know which ones are available in the US).
I don't know if ATF would damage the transmission, but I wouldn't risk it. Things like Oring, gasket and synchromesh compatibility can be critical, especially on older stuff.