I see the dynamic OP.
If she's always been more intelligent than you - or at least posited that idea in your head she's worried your DS is getting a 'better' education and will turn out brighter and with better grades than her DD, making her feel inferior to you for the first time in her life and this area appears to be important to her.
it sounds like her ego and identity are very much tied to her image of being 'clever' and she can't let it go for that reason because her DD is her by extension. She's worried her 'value' to herself and the outside world will slide because possibly your DS will do better. (No guarantee of this of course. Plenty of state school kids do very well and work their arses off in difficult conditions and her DD could be one of these. Likewise there are plenty of private schools that don't get great results at all!)
What do you have in your relationship that you are better at?
Sometimes these things are balanced until an divergence like this comes along and unsettles it all.
She may also think in some unarticulated way that you have undermined her by choosing private because it sounds like she mostly teaches in state school. So by sending private you're saying her job/her whole identity isn't good enough. Ouch. That's a core human truth we all feel - not good enough.
Has she watched this documentary
it sort of shows all sides. For the right person, private school is brilliant. For those that don't need any sort of leg up in life, like the young lady who became a doctor going to Cambridge, she was focused from the start and achieved her dreams anyway.
This may polarise her more watching this.
I'd start to big up state education, her DD, saying you think it's a wonderful school, it's a great place to send her DD, that you've only chosen private because your DS is a bit sensitive and may even have special needs or something l like that.
if you read "the power of now workbook" by Eckhart Tolle it really reveals how many of us act angry and threatened when our status and external image feel threatened. Ego death.
This is what's happening to your Dsis. You need to reassure her that you still think she's amazing and wonderful and clever and intelligent and her DD will do amazing at her state school (because she almost certainly will!) and try to reassure her that you love her.
This is a very 'evolved' way of looking at things. Most people will just have a huge argument and never speak again.
Good luck with the journey of growth!
Final tip: and read the comments below for extra learning!