Iclepicklesmum - just a point of view from someone on the receiving end of this - I don't think this is necessarily the case for you at all, but it might be worth thinking about.
My mum started on my primary school about me, at exactly the same stage, over exactly the same thing of moving backwards into grade 2 from a grade 1/2. Aged 6 I had a reading age of somewhere in the mid teens, could do algebra, was doing grade 7 piano, blah blah. But socially I was very much a six year old, and not the world's most insightful one at that.
The school were very very nice about it. They said no, because socially and biologically i was still very obviously six.
But my mum went on and on and on about it for years. I think it was damaging in that it gave me an academic-princess complex by making me focus for many years on the fact I was so far ahead academically, while being completely unaware that this socially- vacuous princessiness made me deeply unpleasant to my peers.
It would also have come from the fact that my parents don't do empathy in any way, so after some time out in the real world I am just beginning to realise how odd my upbringing was.
What would have been really very good for me would have been to learn at that age that i should see every other child (and teacher, librarian, secretary and cleaner) in that school as another human, with a life and challenges to overcome, and interesting insights. Instead - with encouragement from my parents - I saw the kids as thick and boring, their parents as uninsightful, the teachers as unimaginative boring jobsworths, and I never even considered the existence of the librarian or cleaners.
Not sure what would have made me develop such empathy (other than 20 years in the real world). Scouts or sunday school or something might have helped, but my parents didn't understand the value of doing shared activities - they just said "oh you already know how to do all the stuff you'd get badges for anyway, what would you do that for, you'd be bored".
Anyway - I don't think you would necessarily go on about it etc., and i'm sure your daughter is empathetic beyond her years as well as academically gifted. But it's worth thinking about these things just a little bit, maybe.