There's a service bulletin on the diff issue. Search for it. It's a simple diff fluid swap at the dealer.
2011 E93 - Space Gray - 6MT/Silver Novillo/
OE Tuning/Gruppe M Intake/ACS Springs/Vorsteiner CF Type I Rear Diffuser/Textured Matte Black Custom Powdercoated OEM 359s/Painted Reflectors/Matte Grille, Hood Vents and Side Markers/Arqray Race Exhaust/MS Pulley