It would be a real challenge. Good luck!
Many of our mountain passes get over 20 feet of snow annually. The buildup from the snowplows create vertical walls on the edge of the roads that are 15+ feet high. The Cascades can be a real challenge in the winter, even in a 4 wheel drive truck! ODOT uses vulcanic cinders to "sand" the roads. A layer of this builds up on the shoulders on many of the state's roads in the winter. This pumice- like abrasive is really hard on bikes. It also makes it very difficult to tell where the shoulder is located.
I aborted a tour in March on Highway 26, near the base of Mt Hood. The road had a 12' perpendicular wall of snow on the shoulder, it was melting onto the road, getting dark, and the ski areas had just closed for the day. I calculated the odds of making it to Portland, about 50 miles away and the end of my tour, in one piece. I didn't like the odds so I called my wife, and told her that I would buy her dinner if she picked me up.
I can't imagine Washington's moutain passes being any better.
Taken on July 4th