Remember that no chosen path offers any guarantees.
A child may thrive at either a state or private school. They might also academically struggle or "fail" at either place. So many factors come into play when it comes to "focusing on education": educational needs (dyslexia, dyspraxia, ADHD, etc.), how they get on with socialising, anxiety levels, interests, etc. There are no guarantees that a private school will really bring any substantial, meaningful advantage to your child's life, because she is her own person. She might go on to make a living as an artist at craft fairs, or find her passion doing bar work, or end up in a career via an unconventional route, or be unable to work, or any number of things, that would render a private school placement rather irrelevant. OTOH, it could be of huge benefit to her if it happens to be a specific school that is a good match for her, she likes it and gets supported for her academic needs. No guarantees either way.
Similarly, you might have another child and they end up becoming best of friends, bonded and thrive in the sibling relationship. Most likely is that they will have their normal sibling ups and downs but enjoy having a peer to experience family life with. OTOH, they might end up hating each other or simply not getting along, or not being interested in each other. Another baby could have huge additional needs or disabilities, taking up all of your time and changing the course of all of your lives.
I'm stating a lot of obvious things, but I'm doing so to illustrate fully the fact that trying to balance decisions like this on the basis of life plans is essentially pointless. It comes down to what you want.
Do you want another child? If so, go for it. There's not point deciding to hold back from adding to your family in order to pursue private school, assuming it will be amazingly beneficial for your existing child. It might not. She might hate it. But if you don't really want another child and would be doing so only to provide a sibling, then don't. Enjoy the fact that you can then spend more on education and activities for your existing child.
Basically, there are no guarantees at all, and life can't be planned out as if there are. A decision about whether to bring another child in the world can't be made based on budgeting and logic alone. Of course it's essential to consider real world practicalities, but nothing is promised, anything can happen, therefore, decide to have a baby based purely on what you deeply want. Nothing else. The desire to have or not have a baby is what makes everything that comes after worth it, whichever path you decide.