According to your description, the south room should be your master bedroom, the south is a fire, and the balcony of the south room is connected with the living room, so the sun is too strong, which is not conducive to the health of the owner. If the bay window in the master bedroom is replaced by a sliding door, as you said, the thermal conductivity of the glass is higher than that of the wall, and the radiation area is too large, which will undoubtedly make things worse, leading to discord between husband and wife and hurting the owner's life.
If you don't want to do that, you can also use double or three layers of insulated colored glass, and the door frame can be made of aluminum-plastic materials. But this is the last resort.
Please think twice.