The diff bearings in my E36 LSD went south a few years ago. I had mine rebuilt because it was an LSD and in good shape overall, but for a catastrophic failure like yours, I'd probably pop in a used one.
I also use Severe Gear since it seems highly touted on BobIsTheOilGuy but in 75w140 grade since I have an LSD. Personally, I would not push a car in cold weather at high speeds 2 minutes after start up. My drive takes me more than 5 minutes in the hood before I get onto the 401, which I drive gingerly. By the time I'm on the 401, the fluids are warmed up and flowing well. I've seen a demo of cold diff fluid and it does not seem to flow very well at low temps (google Lucas diff additive video). If I were you, I'd change my daily cold weather drive and get onto the 407 at the next on-ramp.