I don't think they can afford to have another dyslexic child in the class. - aaaahrgghhhhhh. No. This is not the problem. It costs them nothing to have another dyslexic child in the class. Absolutely nothing.
They can't refer him to an EP to get him assessed for dyslexia because they have very, very, very limited time with an EP. And they can't tell you to go privately because it's a state school. (I think - mrz, can they advise you to go privately?)
But if you pay privately for an EP report that says he's dyslexic it costs the school nothing.
What will they do for him in class if you pay for an EP report? It varies from absolutely nothing. To a tiny bit of extra time with the class TA (doing toe by toe or something like that) - which doesn't cost them anything because they're already paying for the TA. Or maybe they'll laminate him something with reminders on it.
What on earth do you think they would do differently if they had a piece of paper saying he's dyslexic, got very poor memory skills, has no concept of time, writes his numbers backwards, and can only spell phonetically?
What would you want them to do?
Schools don't believe there is a cure for dyslexia. All schools do for dyslexia is get you a scribe in exams if you can't write, tell you it doesn't matter if you can't spell, get you extra time, allow you a computer.....
What they don't do is improve his dyslexia.
Only you can do that.