Closet walls pulling away from structure

Until you tear it out and see what you have to work with it is hard to know what will work to replace it. Once you open it up I would find some type of reflective insulation to try to keep the heat out and install that first. If you have vertical supports running up and down the front wall you may be able to screw cedar boards directly to the structure. You would want to cut the edges of the boards at angles or rabbit them so they overlap to form a seal. If it gets that warm there you will want to make sure the boards are very dry before installing them. If not when they dry out you will have gaps. I would probably put a piece of thin plywood behind the cedar to act as the seal.