It is hard to find information on it, but I was told by a dealer mechanic that cars with the BMW M Performance limited slip differentials should use:
BMW Differential Gear Oil SAF-XJ+FM part number83222282583. He said that although for some clutch differentials the BMW specification calls for their standard differential oil, there is a service bulletin out basically stating that if clutch type limited slip differentials are making noise to switch them to the BMW differential oil with friction modifiers.
Based on Forum consensus, this is the same as Castrol Syntrax Limited Slip 75W-140
Both of these oils contain friction modifiers for the clutches.