While chains would help, putting them on and off more than a few times would get old very quickly.

Have you considered performance ice radials instead of your Michelins (which sound like they might be MXV4 all seasons)?

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2007 X3 3.0si, 6 MT, Premium, White

Retired:
2008 535i, 6 MT, M Sport, Premium, Space Grey
2003 X5 3.0 Steptronic, Premium, Titanium Silver
2002 325xi 5 MT, Steel Grey
2004 Z4 3.0 Premium, Sport, SMG, Maldives Blue