It's amazing how well this all makes sense if you assume Klaus is gaming the system. He wouldn't be of much value to her if he lost his faculties or position, so she has to give him a lot of latitude to keep up appearances, but she assumes she has total control, just like she assumed with Agatha.

Meanwhile, Klaus has arranged it so that now just about every military threat to the Empire is caught between two rings of walls and a zeppelin ceiling, ensuring that Mechanicsburg is both contained and protected as well. And he's managed a way to contain his loose-cannon son and incite a plausible excuse to develop a revenant cure without betraying his own vulnerable status, and now he's using Agatha to keep his son off balance and perhaps even feed him enough clues to communicate the situation to his son. Perhaps he's even adding just enough erratic behavior that his defeat at the hands of Gil, combined with the story he fed Phil at the hospital, would solidify Gil's status as the Storm King.

You gotta give it to the Baron: the man's good.