I think this is a great way to go about it – it gives meaning to cold weather without making it too punishing, and gives the player a kind of built-in motivational timeline to get “proper” beds (by which I mean 1 per hearthling, not necessarily higher level ones) organised.
Most kingdoms shouldn’t have trouble getting tier 2 beds by their first winter; but in an emergency any bed + brazier combo can be used in a dormitory style setting in order to quickly protect the hearthlings from the cold. So there are multiple options to solve the challenge and it’s not a “whoops you missed a deadline and now have a cascading penalty” situation.