As a former marketing manager in an Independent school, the stock answer would be "Upon acceptance into the school, you signed a piece of paper which agrees to the school's use of photos and information for publicity purposes."
On results day, I would always use the names & grades of the top 5 high achievers but ONLY if they'd got into their first choice. In your case, I would have phoned first / collared you at school to see if I could mention the grade success whilst carefully avoiding the Cambridge issue.
One year we had triplets taking A levels. One had stunning success, one missed out completely and one achieved her first choice.
I didn't put them on the website as I thought that focussing on one success might upset the sibling who missed out completely. However, I met the mother the following week at GCSE day & she asked why I hadn't mentioned her uber successful child. I explained & she said it wasn't a problem, so I revised the listing.
I am now, as a mum, going through this with DD who got stunning A level results but just missed out on her first choice by a whisper. The school has listed her as a high achiever, but in DD's eyes, she feels that she's got nothing to celebrate.
OP - speak to the school. If they won't remove her name, then at least tell them to stop publishing the negative aspect and to focus on the positive - 4 A* - bloody brilliant!!!!!!!!!!!!!!!!
for both of you. It's a strange week, isn't it?