Sorry, posted in something of a hurry before.
I think that you need to take a long look at your DS. Do you think that it is only in this particular peer group that he shows this type of behaviour? Is he different in other environments - e.g., at home, at Beavers, at football, in the park with other friends? Is he entirely different during the school holidays? Or is his behaviour consistent(ish) in all environments?
The thing is, if you move your DS to the private school, you will change his peer group and his environment - and if it is only in his current context that his behaviour is a concern, then you are right in thinking that it could be a positive move.
However, if his behaviour is consistent in all environments, then the move is unlikely to solve the problem, and private schools are very intolerant of behavioural difficulties as it impacts on their fee earning potential. You would be better off moving to another state school where behaviour management may be better but also suitable interventions would be available.
Also, you do not say whether the SEN the school is investigating is linked to his behaviour, or whether it is something separate? I would be very nervous about sending a child with any degree of SEN to a private school UNLESS this is a known specialism of the school. Again, private schools rely on their 'output results' to pull in future punters and they are intolerant of those who pull the averages down.
I suppose what I'm saying is, ONLY make this particular move IF your DS shows this behaviour only in school and only in the context of a particular peer group and if his SEN is very minor or you think only investigated because of his current behaviour (e.g. my DS was on the SEN register as a selective mute.. which was school induced. He moved school, mutism stopped). Otherwise, a move to another state school would be likely to have a more positive impact.
Established jobshares in primary school, by the way, can be extremely positive, and children often thrive through having two teachers with slightly different 'slants'. However, I can understand that a hastily-arranged 'forced marriage' type of jobshare under difficult conditions for the school may be less positive. I think tbh that the main concern is not the jobshare per se, it is the level of change and disruption to the class through several changes of staffing during a year.