Not an expert either, but I did have a cochin and a silkie last winter. They did fine, although last year was very mild. I did learn all my chickens hated snow and would avoid it if possible.
Maybe you could cover part of the run? My run is fairly small (4' across) so I made a cover for it using an old shower curtain (translucent to still allow light), pvc pipe, and
pvc pipe clamps. I attached it to the run using zip ties. Held up just fine.
My favorite memory from last winter was when I let the chickens out after the first snow. The barred rock didn't care and walked through the snow to get up by the house where it had melted. My cochin and gold sex-link refused to sully their feet and flew across the snow up to the house. The silkie, not to be left out, flapped her wings and took a majestic leap into the air to follow and face planted into a snow drift. (Then ran up to the house complaining loudly). I probably shouldn't have laughed, but...