I really don't think it's that convoluted. Everyone except Thais can see that it is only Odilo's old-fashioned gallantry that is making him refuse the blanket he badly needs. Severian's "why don't you use it until I'm ready for it?" is transparently a way of getting him to take it without losing face.
Thais doesn't say anything until Severian has said this, so most likely she was about to say "I could do with another blanket, if it's going spare."
Odilo would never have accepted the blanket after that, so the compassionate Pega, who has already taken a shine to him, quickly shuts Thais up.