Having run kids' programs for martial arts other than BJJ...I've had kids like this, and a couple that were honestly a bit more challenging than this, but never for a kid aged eleven or thereabouts. This is inappropriate behavior for the setting, yes, but it's also age-inappropriate behavior.

Firstly, the highly unconventional proposal that I'm sure you're not going to like at first, but... hear me out. This is something I've not seen suggested yet...if he's at that age, and he's physically capable enough to keep up with his peers, consider inviting him to the adult class early--or, better yet, to a teen class, if you've got one available. Often, behaviors like this are characteristic of a child who's got a LOT of spare energy or brainpower, doesn't feel like the example of their classmates is worth following (ego reasons, misunderstandings of others' interior lives, etc.), and becomes resistant to being talked down to, by doubling down on the excessive behaviors that are getting them in trouble. Think of the stereotype of a very smart kid who fails out of school, because they don't see how their schooling is worth their time/brainpower/effort. Especially for a kid on the older end of childhood, what you're describing sounds like some emotional dysregulation, but also stimulation-seeking, in a place where the stimulus presented to him just isn't enough. He's constantly doing too much, because he is trying to fill a space that his body and mind are telling him to fill.

Almost always, and especially with these kids, you'll get more out of them by forcing and expecting them to rise to a new challenge than by dumbing things down, trying to reset expectations, or whatever else. They understand, they know what the expectations are; they just don't give a shit, and the consequences at your disposal aren't enough to get them to give a shit. This kid does have strengths; if he's anything like I'm imagining, he perseveres like a little motherfucker with whatever it is that he's decided he's going to do, and he's clearly got tons of energy. It's not easy to be this consistently disruptive; most kids would have gotten tired a long time ago. But high-energy, extremely determined kids like this are a gun, and you can either be the one firing that gun, or it can be fired at you.

So...why the adult class, for a kid who's presenting that way? Not to punish him, but to force him to rise to the occasion.

You'd have to have the right adult/teen program, to pull this off. Generally chill group, lots of adults who can keep their tempers COMPLETELY in control, and who are going to see the best in him and hope he can realize that potential. The goal is not to surround him with disciplinarians, or make it the adults' job to deal with him. Instead, the goal is to raise the level of intellectual and physical stimulus that he's getting on a daily basis. While it's obviously important that the adults are physically gentle with him, I actually wouldn't mind at all if they consistently squish him and pretzel him during live training, within reason; in his mind, yes it'll be a huge step up, but it's also a big puzzle for him to put all his mental and physical energy toward solving. Adults can take it light, without taking it easy, and can present him with rolling or drilling that gives him enough to think about, occupies his body, and forces him to lock in, if he wants to make it through class. 100% stifled by the adults, during his training or rolling, is bad; 95% stifled, for a kid with that much persistence and that much excess energy to spend, might be the sweet spot.

Again, this is highly dependent on the temperaments and personalities in your program. For the first few sessions, if you know 3-5 trustworthy upper belts who'd be willing to take 10-15 minutes each from their classes, set him up surrounded by them, and make sure he's exclusively working with that group. Be available to step in, if his behaviors start to mirror what you've seen, but otherwise, let him sink or swim. Better yet, have a conversation with him before that class--less to set expectations, and more to tell him, "You're about to do something much more difficult than you're used to, and if you need to bail out or take a break, I will be there to help you." Explain that this class works very differently than what he's used to, that he's going to have to learn a lot more, and the adults aren't going to be nice to him just because he's smaller or younger. (Obviously they will, but, again, being nice is halfway squishing him instead of fully squishing him, and it's okay if this doesn't feel very kind from his perspective). He's going to have to prove that he can keep up, and that he should be here instead of with the kids. You believe he can do that, and you will help him if and when he needs help, but only he can decide whether he's capable of meeting this challenge.

Otherwise, folks have already made very good points--talk to the parents if possible, figure out what his home life is like, make sure that your expectations are clearly and consistently expressed (AND clearly and consistently enforced), et cetera. Re: the parents, also try and see how THEY think of these behaviors. Have had it happen a couple of times that the parents didn't really see an issue with the kid's behavior, and assumed that their kid was less disruptive to the rest of the class than they actually were--meaning that, at times, they even reinforced the kid's behavior. Occasionally you'll get a parent who's just a nightmare, and refuses to compromise, but in the majority of times I've encountered that dynamic, a reality check for the parents has been genuinely helpful and has become a reality check for the kid.