Bruce is a naturally protective person. This... pretty much plays into every aspect of his life, public and private. Despite the fact that his!Zatanna has royally screwed him over in a way he finds an irrevocable breach of everything it means for him to trust someone, it was almost too easy to sidestep into the role of 'protective teamdad' with bb!Zee. Given how many various/alternate versions he's met of all the people he knows it's usually not too difficult for him to compartmentalize his feelings and just take them at face value.

Which is pretty much what he's done with bb!Zee.

Plus, he pretty much always carries around a feeling of debt to John no matter the universe so he's determined to look out for her despite her best efforts to make that impossible.

He definitely doesn't want to step in and be a surrogate dad (GOD DOES HE HAVE ENOUGH CHILDREN ALREADY OR WHAT) because that would just not fly with Gio's memory but he wouldn't mind being like. Surrogate uncle or something.

Also she's adorable and he wishes she'd take care of herself instead of being stupid and reckless LET ME TELL U 'BOUT MY STUPID RECKLESS CHILDREN, ZEE. THEY GET CROWBARRED. no jason I love you come back

He would also really like it if she could not grow up to screw with his mind/memories please. At all. Ever.