Sorry I can't help with info about the schools.
My feelings would be the same as yours about tutoring such young children but if they have to sit test papers/undertake tasks would it be worth making sure he has had a little bit of practice at similar sorts of things so he isn't 'thrown' by it on the day.
Also sounds like you might need to find some way of convincing him to concentrate & complete the assessments
Actually from what you say perhaps they will be hard enough to interest him.
Another point though - I would have firm words with his current teacher and make sure he is given appropriate work for his level of ability. The teacher should be planning as many different levels of work as necessary for the range of pupils in the class. From what you have said this is not happening.