I don't think you could make a GDPR request to a private school as what would you gain as your dc would certainly not be able to attend. Private schools can admit who they want and reject who they want too.
Re fit, I would agree with that, I have a ds at Hampton and whilst the academics are great he would have perhaps been in a higher level sports team and in a choir or participate in a concert at a less academic school where more participation is encouraged whatever your ability.