I live in chicago and had that problem with my subaru sti but it was more a clearance issue in the front. My garage opened into an alleyway and there would be real deep ruts in the alley. I think you will have a bigger problem with the plastic shield just under the front lip of the car rather than the diff. Otherwise I drove my m3 all winter except during the storm when we got 21 inches. I had no problems. Granted I don't have to deal with those ruts in the alleyway anymore either though.
2010 X5M - sold
2011 E90 M3 - AW, FR, ZCP
2014 550i xdrive Carbon black
2017 f80 YMB, SS, ZCP on order for ED!