My DS is dyslexic too, OP.
A few things which might be of practical benefit. One is - are you relying on the school's assessment of his dyslexia, or have you had a private ed psych report done? I recommend the latter, because as well as giving a full breakdown of his dyslexic traits (working memory, processing speed, etc.), they will also give you an assessment of his overall cognitive performance. My DS is up in the high 90s for cognitive ability and down in the low 10s for some of the reading-specific tests! It is useful to have an idea of just how big a mismatch between cognitive abilities and reading/writing disabilities you're dealing with here.
Second - if you want to stay in the private sector, check around for the best SEN provision. One of the big private schools round my way has an excellent SEN department, and in fact many parents send children with dyslexia there to get specialist support. I think this is quite unusual in the private sector (most want "easy wins" to keep their GCSE and A level grades high, to keep attracting students), but this type of school does exist. We thought seriously about pooling all the family money to send DS there, but in the end he was unhappy with leaving an established friendship group, so we kept him where he was, and he's now at an excellent state secondary, again with good SEN support.
Finally CAT scores are funny things. DS bombed his SATs, because the primary would not give him extra time (at primary, it's at the discretion of the school) even though his ed psych had recommended it. I hate the fact that these are then used to predict target grades at GCSE - fortunately at a half-way decent school teachers know the shortcomings of these scores and will take them with a large pinch of salt.
If you can lay hands on a copy, today's Times has an excellent article on people with dyslexia who've succeeded in their chosen career path, pointing out that it's one of these cognitive differences which actually comes with a host of "strengths" (although everyone normally thinks in terms of the problems it causes). Often people with dyslexia are very gifted at visual things, whether that's art and design (Norman Foster is dyslexic) or science (astrophysicist Maggie Aderin Pocock is one of the people profiled in today's Times article). They often end up developing excellent memory skills, because they have to! (DS's piano teacher is always amazed at his ability to memorise music).