I would definitely move him if I were you, OP, but please don't panic.
This is fairly typical of an intensely - or even fairly - academic independent school. They manage out the lower, average ability children to keep their stats high. They will then cream off the best of the state school crop at 11+ to keep their stats high. It is all about keeping their stats high. At teacher level, of course they also want the children to be happy and thriving (and to be fair highly academic children can genuinely thrive and be happy in a highly pressurised academic environment), but the results are how the schools keep their businesses going, and they have to protect them at all costs.
We are in SW London and I would absolutely agree with comments about the competitiveness and parental ambition displayed by many - not all, sure, but many - parents at these schools. Of course there are lots of rational sensible parents BUT some of the stuff I've heard about in the London/home counties area is insane.
But none of this means that your DS can be 'written off' in Y5! Neither is he 'condemned' to 'just' getting Cs. Not that there's anything wrong with Cs, but it is really, really common for DC with additional needs to come out with lower scores around this age, and it doesn't mean that any "predictions" will automatically hold. With the right support, DC can make amazing progress in secondary.
My DS1 is on the autistic spectrum and has severe attention deficit disorder. He didn't do CAT scores as was at state school but I can totally imagine him being briskly managed out had we gone down the prep route. Instead he went to our local state secondary and eventually came out with all 7s, 8s and 9s at GCSE - way above Y6 predictions. As did many of his friends who have similar SEN diagnoses. They really hit their stride - with the right support - around y8/9.
What you are being told is that the school your DS is in is not the right school for him, which is probably true, but that doesn't mean he isn't going to flourish somewhere else.