The hay is good because it keeps the chickens from destroying the roots, causing a nitrogen buildup (nitrogen burn), and hardening the soil, which would kill all growth. Every time you get heavy snow, shovel it and put carbon down. Seeds will also go down into the ground, what is not eaten by the chickens, that is. Keep in mind that the hay will act as an insulator for the ice when the melting season comes, sort of like ice blocks coated in sawdust for freezers in the time before refrigerators.
I am constantly changing the setup. I do not know if I could commit to keeping it the same. For frozen eggs I just collect them more frequently, but it sounds like you are unable to do that. I have seen people insulate the nests.