Bigby is ancient, far from human and a tiny bit incomprehensible even to other Fables, let alone anything younger than that.

"Bufkin would get arrested inside ten minutes. It's nothing to do with what he is and everything to do with the fact that he's a menace." He gestures vaguely with his hand. "Keep it. At least until we're inside. I don't need it. Just people expect you to wear a coat out in this. And I'm not giving Gren an excuse to get on me for treating his people badly."

Sure, that's why he'll escort her back.