There are lots of things to look at. Firstly, if the "connections" issue feels morally wrong to you, then I really think you do need to think twice. It is what many people want from an expensive education. I can say my DD has benefitted immensely from connections she has made, but it has not been via her individual school. I am delighted because we have no connections whatsoever! Yes, the boys' school networks are stronger but the girls can tap in! In the future, as more girls break into politics, the City, Law, etc girls' schools will become more effective at this. Going to a top independent school also tends to colour who your friends and networking buddies will be through university and beyond. It is often these people, and their parents, who help you out.
You definitely cannot say a bright child will necesarily do better in an independent school. I also feel that some independent schools do want a certain type of conformist child and ones that stand out from the crowd are less well accommodated than you might imagine. Some schools have a certain "type" of child that fits their model. You need to ascertain if a child who pushes herself forward is the type for a nurturing school. She may need a school that allows her personality to flourish but encourages a more collaborative approach to learning.
I always think there is a worry when one child want something that will cause the family to have less disposable income than it otherwise would have had. Although your DS is not interested, at the moment, who is to say he will not add up the sums you have spent on DD in the future and wonder why he did not have £X spent on him. He may not, as a very young person, have wanted it, but he may in 10 years time, be less generous towards you. Did you ever take him to look round an independent school when he was 10 years old? Or do you just favour the child who, by the sound of it, is very up-front with her opinions and lets you know exactly what she wants? Are you happy that the family will have to scrimp and save and that will inevitably further impinge on your DS? What will you say to him now and in the future when he cannot have something or when you have insufficient funds to pay for a school trip for him or his university top-up living costs? I would not be putting myself in your position and I think I would look to move for A levels only - and offer this opportunity to both your children.
If the state school has a good 6th form, you are in a very good position. University entrance is about results and applying for the best university you can get into. Independent schools are good at that, but you really can do the university research yourself and the private school may not have any advantage over a very good state school. You can aim high from whatever school she goes to so long as the teaching and results are good. I would treat them equally, save the money you would have spent, and spend it fairly on both your children, not just the one who is asking for it.