My title is a bit of a mouthful but wanted to get all the key info in there!
DS, currently 11 and in year 7, has an ‘interesting’ profile. He has been diagnosed with ADHD and is very intelligent. His state school doesn’t have sets and they report that he routinely finishes his work (correctly) much before other students. If not immediately engaged with other work or - in his least favourite subjects, in the class in general - he can be chatty or fidgety and at times, provocatively and actively disruptive. 
The SLT love him and are really doing their best to support him. The SENDCO said they’ve been failing him by not challenging him enough. They give him extension work and are actively encouraging his natural talent in two specific areas.
However, the underlying issue is that he is not sufficiently challenged and engaged in class when in a classroom, Covid permitting. I think the solution may be moving him to a selective-ish independent with smaller class sizes for individual attention and a rigorous pace. We’d move him for Year 9.
At least half of his best friends are girls so we’re definitely only considering co-Ed. We can do central London and the least far out parts of both SE & SW London. As he’s hyperactive, he needs somewhere with good outdoor space and sports. I know this limits our options probably to Alleyn’s & Emmanuel. Is there anywhere else?
He’s always been in the state system so I have no real understanding of 13+ and am aware he may be at a disadvantage in terms of preparation not being in an independent.