My partner, who makes that trip a lot during winter, responds:

The bike lane on the bridge deck is eventually cleared, but usually not the first day or two after a snowfall. I check the hwy 1 ironworkers webcams, you can usually get an idea of whether it has been cleared by looking at them. On the north shore, the district does not clear bike paths quickly but the city is very good. He can look up the boundaries and plan a route that keeps him mostly in the city.

My opinion: The bridge gets too dicey for my comfort a few days every winter, especially because of the long northbound downhill decent followed by a poor sight line curve. It frosts over much faster than any bike lane on the ground. I recommend putting the bike on a bus to get over the bridge on days when it has snowed recently or if it was wet the day before and temperatures have dropped below 0.