I had a class in which I was required to grade myself, share the grades with the others in the group, and then defend or argue the grade that I assigned myself.
Big difference: I was a senior in college, about to get my degree in elementary education, and this was a class about grading, how to deal with a student/parent who might argue about a grade I gave, and how grades can be sometimes arbitrary or influenced by certain factors when the question requires something other than "black and white" answers (like when the question is "what is the capital of New York" vs "what is the best city in New York"). And, I knew what this class was about and voluntarily took it. It was not required, but it was interesting as an elective. As part of the class, we discussed certain anxieties or nervousness about defending or assigning grades, and we also learned about boundaries and privacy and sensitivity.
Your daughter is nowhere near a senior in college. She's young, inexperienced, and immature as far as the education process goes. (Not saying immature as an insult or disparaging remark, but just saying that ninth graders are still young, still having just turned into teenagers, and just starting high school, and it can be a rough time).
I do think this is a FERPA violation. FERPA, as far as I understand, requires a parent's or student's signature before releasing personal info, unless requested by law enforcement officers, or if the student is applying for some scholarship or acceptance into a school and knows that his/her grades will be released to the appropriate committee, or to an authorized state accounting agency for enrollment data purposes, etc.
As far as I know, the Buckley Amendment is the part of FERPA that protects the privacy of a student's information (grades, address, discipline records, etc) so I think you're talking about the specific part of FERPA that is Buckley, not other aspects of FERPA. (I could be wrong).
A couple of questions that could be pertinent: is this school that your daughter is currently in a public school or private school? If it's private, they could argue that they are exempt from FERPA, as they don't receive federal funding. Next question: does your daughter receive any therapy or counseling? A medical doctor who treats her and realizes that she experiences anxiety and social panic could issue a statement that protects her from having to share this private information.
I feel that the most important parts of your question are: your daughter's clearly stated anxiety over this loss of privacy, and the fact that you are talking about ninth graders. They may be 14 years old as an average, if I'm correct, and this is a very vulnerable age for bullying, teasing, embarrassment, shyness, ridiculous bragging, unrealistic expectations, unhealthy comparisons among peers, inappropriate internet sharing, and other difficult situations.
Can you imagine one child standing up and revealing a BMI and a grade in public, and then having some other kid posting that on the internet (especially one of those sites where the post or pic disappears soon after its read)? Does the school realize the implications? What if someone went to the school administration and asked something like "my kid got a D in history, and I think that it's unfair. I want to know what the other kids got in that class"? The school would hopefully refuse that request. But they're letting teachers do it without any parental guidance or school staff oversight?
I encourage your daughter to advocate for her own privacy, and I encourage you to tell her you will support her. Write a statement to the teacher together. Declare that in this day and age where anything and everything ends up online, and where bullying is a constant problem, that your daughter will not reveal her private facts, nor will she listen to others' revealing theirs. Have her inform the school that she has not given her written permission for the release of this sensitive information. If it's a private school and if she signed something in the syllabus and knew about this beforehand, it might be a more difficult fight and she might not have the law to rely on. But that doesn't mean she can't fight for something that is important. If she fails the class, have the teacher put the reason why in writing.
Make sure your daughter understands that this isn't about disagreeing with a teacher, or disagreeing with a particular assignment, or simply not liking an assignment. Make sure she is very clear that this is not something arbitrary or trivial. Make sure she knows that if it were a situation where she didn't want to write about something "boring" or work with "student X who's such a pain" that there would be no excuses, no defense. This is important. I hope she doesn't just say "peace out, suckas". I hope she writes a memorable statement about privacy and sensitivity and respect for others.