If it were me I would want a seperate winter bike set up for days when the weather is really bad. The warmer season commuter bike is good in the cold as long as it is not snowing or freezing rain or the roads covered with ice and slush or snow.
I would get an inexpensive used mountain bike with rigid front fork or a cheap dual suspension WAL Mart bike and put aggressive studded snow tires on it . The widest it can fit. And also put big off road fenders front and back. The off road fenders for mountain bikes sit further away from the tire and won't clog up as easy with slush. If you can find a cheap frame on e-bay that can accept disc brakes that would be even better as they work somewhat better when wet than rim brakes.
This way you don't have to worry about if your prepared when the weather is bad. Places to look for really cheap used bikes are garage and yard sales and on Craig's list.