How controlling to not bring outside clothes for her little one. Does she really think she can dictate your daycare activities? I get that a one year old will probably be in a wagon/stroller so they do not generate body heat through walking. That is why I bundle up my non walkers with a blanket and if it really is too cold we do stay in or limit our time out, but kids need to get outside in the fresh air and the change of scenery is good for everyone.
If a chat about this does not change her mind, then I would advertise for a replacement. How is she going to be in the heat of summer? Does she not think you have common sense?