I would say private. She will go into Yr 7, i.e. with everybody starting secondary school so socially it will be very good for her - a chance to form friendship groups from the very beginning. I would argue that going into Yr 8 as a very young August child would be a lot more difficult - friendship groups would have formed, dynamics established, etc. It would take her a while to even figure out what's what socially/friendship wise speaking. Also, friends from 5 yrs ago aren't necessarily going to be friends now and, in a sense, her thinking that they will be and then potentially realizing that their paths diverged could be even more challenging than just forming new friendships in a fresh peer group in Yr 7.
I personally think putting an August born child not zooming ahead academically into Yr 7 is a great idea. She will have time to mature by the time of her GCSEs etc and without the pressure. She will not stick out from her peer group physically and emotionally since there won't be much age difference really between her and Autumn/Winter born children etc
Re academics, if it's not a particularly selective school, she will not necessarily languish at the bottom of the bottom sets. And at least in a lot of private schools, bottom sets do get a lot of support instead of being written off as "low achievers" and consigned to BTecs instead of proper GCSEs in some state (generalization, I know, but still...).
If it's a really selective school and she walked in without prep, then she is not such a low achiever after all.
Dyslexia is quite a common condition and support depends on particular school rather than private vs state sector as whole. And at secondary level, it's all about support for exams/extra time, support for using laptops,organization skills etc - private schools often do this very well. Presumably, private schools knows about her dyslexia and still offered her the place so it doesn't scare them.