I grew up in a well-to-do area where many children were educated privately. I went to a state primary and an excellent state secondary where there were some children from very wealthy backgrounds, but also many children who came from poorer backgrounds. (I suspect that the catchment is quite different now and that the school is a lot harder to get into, so probably more socially homogenous :( ).
I came out with Oxbridge-level A Level points and almost a full house of A grades (No stars back then!), but while the teaching was very strong, getting to that point required a lot of my own initiative beyond that provided in the classroom, spoon-feeding was non-existent and, in the lower years of the school, there was definitely a culture where it was not cool to be clever, or get good marks. So getting to those final grades was probably harder than it would have been had I been sent to one of the many 'naice' girls' schools in the local area. I also completely lack the inbuilt workplace/social confidence that seems to come with private education and only really developed it in my late twenties/early thirties and particularly since becoming a parent and, basically, giving less of a fuck!
However, I do remember it being a bit of a minefield in the teen years and friends' parents making an issue of where people went to school. For example, polite or not so polite suprise when I revealed that I went to the comprehensive. A friend, who left my school and went to a small and not-very-academic private school, commenting that people from her school regarded mine as 'rough' and that the families she babysat for were doubtful about having me babysit as I came from a state school. 
It also cut both ways: I blush to remember teenage me sitting in the back of someone's car, surrounded by privately educated acquaintances, confidently commenting that I didn't see why people needed to pay for education as surely you could work hard and get A's at a state school, just like I could...
. To make matters more complicated, I also had a weekend job at a local girls' boarding school where teachers were politely astonished to find out that I, in my catering uniform and rubber gloves, was also doing A Levels and heading for university.
Anyway, my main point is that no-one enjoys people who are indiscreet or loud, so a word in DsD's ear might be useful.
On the other hand, you should also tell your family to unpurse their lips and be supportive of your lovely young stepdaughter and interested in her activities and achievements.