All kingdom's tooltips are hidden when the timer toggles dungeon completion informations (even when we're not on the map).
I agree that a manual toggle would be better. Make "classes 12/18" and "badges 7/10" buttons, and make the mousewheel toggle for extra convenience. Clicking anywhere on the tab to switch between the 2 would be nice (less intuitive but more ergonomic, like the mousewheel thing).
On dungeon selection screen, the tooltip could add more informations, like classes/badges completion. And if you want small tooltips, you can color the dungeon name using difficulty color, and remove the difficulty line.
On the topic: when I hover inventory items, it takes some time to display the associated tooltip: that's irritating. Please, make it instant all the time.
It's the small details that make a great UI.