Driving across the Alps in Winter

In the fall we booked two ski accomodations - 4 days in LaPlagne, France and 4 days in Zermatt, Switzerland. We were in a hurry to book something as resorts were already filling up so we put off figuring the details to now but we did account for a full day of travel. We have booked a flight from our home in Naples Italy to Lyon for the LaPlagne trip. We haven't booked return travel yet but will likely fly from Torino.
The costs of the transfers from Lyon to LaPlagne and the 8 hours of train at 150 Euros a person for our family of four between the two destinations have left us considering whether it would be worth it to rent a car. The cost of the one way rental is comparable to total costs of trains and transfers but we are worried about driving across the Alps in February. My wife and I both grew up with snow in the upper midwest, have driven extensively in New England, Colorado, and Washington state for ski trips and have been living in Naples for the last year and a half - we are experienced snow and European drivers but Alpine travel with a rental car in February still gives us pause. I have read other online forums which suggest we have a pretty good probability of being able to make the trip in a day. Does anyone have any experiences or thoughts on whether this is a viable option or have any better ideas? We have free cancellation on the place in Zermatt until the end of January - if we can't find a good option to cross the Alps, we may look for a good second place to ski on the Western side. Thanks!