disabled = true indeed works (it appears that sometimes I don't seem to trust things to be possible in the simplest way...).
(On a sinde note: It's so weird that $ all too often leads to another error of "Node not found". I stumble over this all the time and I'm sure I'm not alone. Is it because I'm trying to find the same node I attached the script to? Can I only find different nodes?)
What I see now is that even when disabled the mouse-exited still works. I thought that any input to the node should be completely turned off, say, until I turn it on again. (Just as it works fine with CollisionShapes and input_pickable.) Is there any way to do that?