Wow, I am totally gobsmacked at that update 
Though from the POV of school box ticking, I can understand where some of it comes from - only the top 2 kids per year go on the G&T register, which is a joke in itself - I agree with moving schools, but tight budgets could mean without a diagnosis, you could hit similar problems elsewhere.
First off I would consider raising it as a complaint in writing & complain to Ofsted too - also look at the schools previous Ofsteds, you might find some clues in the most recent one to explain this attitude
I want to write a really long post here, but still strugling with migraine, so will maybe come back later.
But important points I did want to raise are..
Based on his GP noticing hypermobility, from things you write, I still wouldn't rule out EDSH as the source of his issues & I would advise not rushing into the Ed Phyc route as it could backfire if there is a physical reason for his issues as there is with our DD. If there is a real officially recognised disability, which EDSH, like AS is , then the school would be breaking the law by not meeting his needs properly.
If symptomatic, which from experience I can tell you isn't always obvious at this young age as its their normal, they don't know to complain & JHMS & EDSH are now recognised by the experts in the feild as the same condition, but it covers a wide spectrum of symptoms - his GP can't officially diagnose JHMS/EDSH, he will need to see a good EDS knowledgable paediatric rheumatologist - but you are in a good spot, as the top experts are all in London.
I've realised with DD that a lot of it was down to not asking the right questions - if she's suffering the "Postural Orthastatic Tachicardia" related Dysautonomia effects, she can be very litteral & only a direct question will get the right answer.
She spent years feeling demoralised by her handwriting skills, which were never below expected levels, just below her ability in everything else - but she never told me her hands hurt if she wrote a lot because she thought everyone's hands did. I felt like crap parent from hell when the OT asked her if her hands hurt if she wrote much - & she said yes & went on to explain how much.
POTS is a symptom of EDSH & a condition in its own right - it's very rare in kids' but it is common in Hypermobile kids & can be trigger - more made worse & more noticeable by an illness, leaving them deconditioned - looking back, in our DDs case this was pneumonia, which left her wiped out & nit recovering well from, later a bad ankle accident put her on crutches for 9 months - things deteriorated from there as she began to hit puberty & more injuries & constant stress of a school not meeting her needs & bullying.
We DID take her out, homeschooled for a few months & moved schools in the end & I regret above all else not doing it years sooner - her next primary were wonderful.
In short, at 7, DD was a bubbly, gregarious, exceptionally bright & mature kid, with a few quirks such as hating seams & labels in socks & being very empathetic to others, she could diagnose others in her class with SN faster than the school did - she was a fidget, & never slept as well as others & could struggle with tiredness as a result, but was very sporty, arty, all round G&T - bar handwriting
She had the same problem with reading that you describe - but I do too, it's not dyslexia, it's brain fog, I get overwhelmed by large chunks of type, it might as well be in Chinese, I thought for DD it was from poor sleep - so I taught her to do as I do & block out the text into manageable chunks with a bookmark or sheet of paper - she was soon reading well ahead of her year & now reads adult books - she was by 8/9
She had the same issue with maths, she works the answers out correctly in her head - she now now also admits the written workings out hurt her hand - in primary this was helped with a seat wedge, writing slope & special pens.
They do need to show the working out to get SATS marks, so it's good to get him to understand that it's important as not everyone is as clever as he is, so they need to see it written to understand (I always found this reverse Phyc stuff to work best with DD or set it as a challenge) - but in the long run that won't matter - he will be streamed according to ability in HS & will be working in a class of others like him, which means teaching us much mire at his level - DD is now loving this.
I hope that lot makes sense
I will sort out some useful links & post later