When my daughter was little, we had a fleece one piece cover up that worked well, that was kind of like
this. It was good for adding warmth without adding significant bulk under the straps (since squishy stuff reduces the effectiveness of the restraints in a collision). If you go that route, I suggest making it on the large size, particularly in terms of length of the sleeves and legs, as that will really extend the useful life of it. Having a zipper that goes down one of the legs definitely makes it easier to get it on child.