With a decent set of winter tires, I don't think you'll be worse off than any other RWD car.
It's not going to be nearly as good as your 328 XI for snow though.
The real downside of the M3 for everyday usage is the gas mileage. But if that's affordable, I would definitely go for it.
2011 BMW E92 335i, 6MT, Le Mans blue - Quaife LSD, JRZ RS Pro, JRZ camber plates, Eibach ERS, full M3 susp, custom toe arms, Powergrid end-links, SS brake lines, brake ducting, Michelin PSS, UUC SSK, ETS FMIC, ER CP, Perf. Exhaust, VAC RSIK, Sparcos; work by VAC Motorsports in Philadelphia