I disagree with the "bright kids do well anywhere" mentality. A bright child will probably get good exam results at a state school. That doesn't mean they have reached their full potential. From experience, a good selective independent school will provide a syllabus which goes way beyond the exam syllabus and stretches all children. When everyone in a class is capable of getting As or A*s you can teach the class to that level and take out groups who are working at higher than that level to do higher level work. The subject choice at independent schools also tends to be wider so, for example, you take 4 or 5 A levels instead of 3 if you're capable of it or continue to take a second language even if not doing an exam in it.
I went to a selective independent girls' school on a scholarship after being very unhappy at a state secondary. I applied myself without my parents knowledge because I wanted to go so much.
I was by no means the brightest at that school. There were some children who were real geniuses and that is a special need which needs specialist teaching and attention in my view which state schools will find it difficult to give. Some of those children may have been fine at state school because they would have taught themselves but some may have coasted or become despondent and demotivated. It was actually very good for me to know that even though I was getting good grades and had been one of the top students at my state school, there were many children who are much brighter than me and it pushed me to achieve.
Obviously there is still bullying at independent schools. I can only speak for my experiences but I found that eccentricities and being very academic were tolerated much better at independent school than at the state school I went to and that behaviour was generally better. At highly selective independent schools there tends to be a culture of working hard and wanting to get good grades so it isn't behaviour which is usually bullied (though the atmosphere at these schools can also be pressured).
I would also say that my own experience was that people didn't generally know how much money other children's parents had and who had bursaries etc. Unless you are talking about one of the prestigious public schools, a lot of children who go to private school are quite ordinary - at mine 10% were on bursaries or substantial scholarships and in many cases fees were paid from inheritance or grandparents and children lived in very ordinary houses and parents had ordinary jobs. There may be more of an expectation that you can afford expensive foreign trips and uniform and so on though.
So I think there's a good chance it will benefit your daughter but it's not an easy decision taking into account your other children and the cost.
Most independent schools have a second hand uniform shop though so you could ask about this.