Despite her wailing, the point of the meeting veered off course and the hostess did not want her guest who traveled all day to suffer. In reality, there was no longer a need to hide her existence as lord Dyne from the world anymore. It helped that even if she stated it, without overwhelming evidence like coming to the castle themselves, they would deny it as impossible.
It was similar to when someone was convinced that the earth was entirely flat, you nodded along and tried to not get mired in the crazy.
Baths were available in suite, and were larger than needed, heat management was always an issue with so many moving parts. So, the baths were huge to have places to put more heated water. Claw foot tubs that could be heated as needed were common luxury items and some nobles like the Elder River welds had inset luxuries baths. None of them however could produce hot water on demand from a tap.
