This shouldn't be a difficult process, really. Bruce is no stranger to adopting children, as his ever-growing family portrait in the Manor's entry hall can attest to. He has no qualms about taking in a child and accepting them wholly and fully as his own. He's not a guy who puts bonds of blood over found family, at all. All of this to say, this shouldn't be something that's got him feeling nervous - but it sort of is?

Yes, he's adopted the vast majority of his children, and he's in no way unwilling to open his home and heart to you as well. It's just... usually his wards come into his home by at least somewhat mutual decision, with some of their own direct input. But in your case, there's a bit of a 'third party' vibe going on here. You're Selina's child, and Selina is his partner, and if he's (finally) convinced her to move in, obviously you come too. And you're welcome! More than welcome.

God why is he feeling nervous about this? Maybe there's just some sort of pressure to do this right. Bruce values family, above most things - heck, possibly above all else, when he gets down to it. And if you're to be a member of the family, he wants you to feel like it. But he also doesn't want to pressure you or overwhelm you by making it all move too fast. He knows how Selina can be, and she raised you - the pure fact that you and he haven't had a ton of interaction and time together so far speaks to at least a certain level of independence and perhaps even uncertainty in accepting a new family, or new stepparent.

He wants to do this right. He has to do this right. For Selina's sake, and his own, and yours. ...He's probably massively overthinking this whole thing, too, isn't he? Damn.

"Hey, kiddo. Morning," Bruce says, keeping his tone casual as he joins you at the breakfast table, the morning after your first night in the Manor. "How'd you sleep? Your room suit you okay?" He stops himself from immediately offering that he and Alfred could rearrange things if the answer was 'no'. Hold your horses, Bruce.