OP, have you considered that your reaction against the school and assessment is essentially the emotion and denial about a possibility that your DS might have SEN, is not perfect as you say?
I mean it in the nicest way, as a mother of DC with ADHD, ASD. I know what it's like, you are having a hard time. Hug and 
Having ADHD, any SEN, has nothing to do with being perfect (nobody is, some apparent 'angels' could be quite troublesome), nothing at all to do with being naughty and unkind, being disliked (you know the speculation that Richard Branson has ADHD?). To the contrary, some diagnostic evidence could involve being perceived 'naughty', disruptive (which your DS clearly is as you complain that bullies at school get less criticism than your DS) because of behaviours due to ADHD which the person cannot help, are not deliberate, are due to a disability. You are aware that ADHD is a disability, not an ASBO? You don't need to be defensive.
Diagnosing, recognising a disability could start to address the problems, to provide reasonable adjustments and to protect the person with disability, which otherwise, without that protection is open to judgement and consequences of being perceived 'naughty', the stigma of being disruptive, antisocial and dysfunctional.
Perfect, naughty, unkind, bully, disliked - all those words you mention do not reflect ADHD, they reflect your insufficient understanding of it and the issues surrounding it (forgive me, but I have to say that. That's how your posts come across to me and this is not helpful for your DS). As a mother of a child with potential SEN you need to clue up, read up on the issues of ADHD, SEN process at school, disability.
Having SEN / disability does not mean being 'bad', 'defective', many children with SEN are very bright and do well in education and in life. It means functioning differently and for this reason requiring the environment to be adjusted to avoid a disadvantage for oneself.
You need to re-examine your own conception of these issues. All this 'upper class' thing is b@&%x. Leafy schools have plenty of ASD, ADHD, dyslexia, dyspraxia and other things. 3 out of 300 is probably those with EHCPs, the overall % of SEN is around 12-15%. Recently somebody told me that Westminster school is a special school in disguise for undiagnosed ASD, Lol. All schools must be trained and do provisions, they do have to find up to £6000 from their existing budget to support your DS and you need to grab and shake them so they see you mean business. It is the job of being a parent of a child with SEN.
If your DS has ADHD, denying it will not eliminate the problem. To the contrary it will snow ball and escalate with the consequences for your DS, not the teachers. This disability would have a pervasive effect on everything in his life. Recognising it might give him the key, the awareness and understanding of how to mitigate it. Hopefully it could provide medication that will not solve everything, but be a huge step forward. Undiagnosed and unrecognised ADHD's impact will increase with age and the dynamic could become quite counterproductive, he may become more troubled and troublesome and fail at school, develop some other problems like addictions, antisocial behaviours, conflicts and worse, if he is left unsupported. There is a very big bad downside in denying diagnosis and medication if your DS indeed has ADHD. Anxiety and ASD as the underlying reason might also be on your radar (anxiety is fight or flight reaction).
I am not a doctor, but from your posts he sounds like a case of ADHD with all the symptoms and criteria. He has continuous daily incidents at school for years. In your own words at home you also struggle with his behaviour, that 'something' daily for years and nothing helps, even though your parenting and environment is fine as evidenced by other children of yours. You sound so stressed and drained emotionally, clearly it's not all zen at home. I was continuously on the edge before my DS got medicated. As it was said above, at home 1:1, in a very structured consistent environment you might underestimate his symptoms.
One possible reason that CAMHS told initially that he does not qualify is based on your own reports and questionnaires. He is your child, he is perfect for you and this is absolutely correct. Symptoms are not value judgements. But being so close to him, you might not realise, might underestimate the severity of symptoms at home and other places. In this thread you said how you struggle with him at home, yet, you don't seem to report that to CAMHS. If you did he would have met the criteria of several settings. If you consider dispassionately many of his behaviours at home and going out, you might recognise it.
You mentioned that he has average ability. How do you know? Do you have a cognitive assessment by an Educational Psychologist? Or do you just mean he has average attainment? What if he is very bright and is underachieving because of ADHD? His behavioural problems will have an effect on his education, even if not ADHD. He meets the definition of SEN anyway..
If the school are referring him to CAMHS and are pushing, this is great for your DS, they are trying to help, he is on SEN radar, I would grab it with both hands. Most parents of children with SEN are fighting with the school to recognise SEN and make provisions. Your Senco comes to you. Take it and start discussing what provisions they are making / could make at school. Ask for his Individual Educational Plan, ask for an Educational Psychology assessment, OT assessment, ASD assessment (?). Get a copy of the SEN code of practice if you don't already have one, talk to SOSSEN and get your plan in order of getting the provisions at school. The school needs to make provisions. Changing schools now will delay assessments, you need to document SEN and diagnose your DS in this school, then you could move. But you need to fight to get medication (if indeed he has ADHD) as it would make huge difference.
I am sorry I am giving you a bit of tough love. Your DS needs you to grapple with that. Try to relax and don't be hard on yourself and others
. SEN is a long term struggle.