op, you can try looking at some of the 'twice exceptional' stuff out of the US, that might help you make your mind up. the yg&t unis sometimes run courses for parents on giftedness and additional needs such as aspergers/ asd. (there was a good one a couple of years ago)
i'm not sure why the school are struggling to cope tbh - it's not uncommon for children to be working some years ahead of their peer group academically, most school s are able to differentiate appropriately without recourse to moving years completely (this has somewhat gone out of fashion after the whole ruth lawrence thing). if the school are really this clueless i might be tempted personally to look at other state school options before considering private.
for example - the ed psych assessment dd2 completed in the summer term of yr r suggested that in some areas she was working at approx 12yo level, and in others only about 7/8 - school looked at moving her to a higher year group, but have decided against. instead they have re-arranged the school timetable so as a whole each class is timetabled the same subject at the same time, so there is greater overall fluidity between year groups - she can move up for some literacy project work - in fact any child can spend some time with a higher year group for maths etc. each of the children have always accessed reading material material appropriately from other year groups, and teachers have been good at stretching children sideways (broadening their horizons with specific music and percussion classes targeted at the brightest children) as well as appropriately differentiating standard curriculum stuff.
there is no reason why a child of high academic ability can not also get along with their classmates (that said, dd2's teacher was aware of the differences between dd2's interactions between her year group and with the older children - but believed it was more important for her to learn how to get along socially with her year group rather than moving her class iyswim). if your dd is having social difficulties, then it's important for school to be addressing those as well as her 'giftedness'.
many gifted children have an IEP drawn up to set out the targets for their attainment during the term/ year. does she have one of these?
these are also a common trend with sn children - so for example, it is not unusual for children to have 2 IEPs running - one for their 'gifted' targets, and one for their sn - so social targets etc. i'm not sure whether you would feel comfortable discussing with the SENCO whether this would be an appropriate way to help your daughter if you are concerned that the school do not know how best to help her?
lots of info on IEPs all over the web - lots of it aimed at sn, but used for all eventualities where children do not quite fit the mainstream norm, or require specific help to reach their potential.
(incidentally - all three of mine were labelled g&t at one time or another lol - i only use dd2 as an example as she is a similar age to your dd and also has issues that require some specialist assistance alongside the 'gifted' label. i'm shocked at the schools inability to cope, if i'm honest.)