Mind control is the One Line that a lot of otherwise very amoral characters in this comic won’t cross. It occurs to me that Klaus doesn’t seem to get this. (And therefore doesn’t get that in their eyes, he’s turning into Lucrezia.)
(And of course: he has more standards than they do, and he doesn’t think it’s a moral event horizon!) But he left the Jaegers out of his “Agatha is the Other” theory because he was sure their loyalty would override their objection to the Other’s MO. And he didn’t take any measures to restrict Bang’s authority or keep her away from Gil—so either he didn’t think she’d notice, or he didn’t think she’d care. In general, or even about Gil in specific.
But then, that’s Klaus’s biggest weakness—he doesn’t reevaluate. He takes time to make his decisions, he gives people one chance, but once he forms an opinion that’s it, mountains will move before he does. And that doesn’t allow the fact that people change.