Hi Emsie, I teach Year 5 and have a whole group of kids like this in my class!
Firstly, if he is very bright and active, as you say, he probably needs some extra stimulus. His teacher should be making sure work is differentiated to challenge him. A lot of the time it's really not recommended to push a kid up a year as they struggle with the social-emotional aspect of being with kids a year older. What you could do though is talk to his teacher about a special independent project that engages him in things he is interested in, and gets him out of his chair sometimes.
Also, something I find useful is having a lesson-by-lesson ratings card, where at the end of every lesson the child rates their behaviour with a number out of 5, or a smiley-face. We focus on cooperation, assertion, responsibility, empathy and self-control so the child sits with the teacher after each lesson and reflects on their behaviour during the lesson. It's quite useful for helping the child to look for patterns and pinpoint triggers for going off-track. It goes home to the parents at the end of the week to be signed, and a lot of the families use it for a reward system.
If he is spending a lot of time being 'talked at' by heads/deputy heads, no wonder he is switching off or acting up more. I've found that boys often respond much better to a bit of motivation or responsibility once they get to that stage. Is there a teacher he has worked really well with in the past, that could do some before-school reading/mentoring with him? Or, if he is a good reader, maybe seeing if there is a way for him to go and read with younger kids occasionally during the Literacy hour? Buddy reading can be a great confidence boost!
The important thing to remember is that you are not failing!!! You are a loving, caring mother who is concerned about her child - let me tell you, that's more than some parents. Ultimately that will pull him through. Boys are very different to girls so don't be surprised that you are going through different things with your son than your daughter. Boys can be tricky sometimes, especially if they don't see much value in what they are learning, but as long as he can see that you and his teachers are on his side, I'm sure he will be absolutely fine.
Apologies for such a long message but I really hope all works out for your family!