I've walked across the Bear Mountain Bridge. It's part of the Appalachian Trail so hikers cross it all the time. There was a pedestrian lane (sidewalk) separated from traffic, if I remember right, but it was partially blocked by construction when I was there. I wasn't thinking about how to cross the bridge by bike, but if it looks too hairy, you could walk your bike in the pedestrian lane (just stay upwind of the hikers).