Well, she's right about that, but wrong that it's merely a localized phenomenon.
It sounds as though your DS might be interested in exploring what he can do without formal accommodations, as well-- so maybe working with someone outside of the school on test-taking and time-management skills in light of the reality of his lower speed at certain tasks would be something that he could do for himself. Do you think his sense of privacy and autonomy is part of the reason he'd prefer to not be identified formally?
BTW, there are privacy expectations that can be written into 504/IEP for individual students. DD also prefers (rather strongly) to never be called out in front of peers-- either for her age, her LOG, or her disability). This is a harder thing for some people to understand and comply with than for others.
If you have a medical/neuropscych reason/label for the issue in play, however, then it becomes a matter of medical privacy, and the law is pretty clear about not revealing those details without the student/parent's explicit permission.
That means that they would be on the hook to provide accommodations that don't identify him.
I'll also add that an IEP is vastly superior to an 504 plan in one other important dimension-- there is a transition planning stage between secondary and college.
You will want to consider what the challenges will be in that post-secondary environment-- exams, note-taking, etc. Both things take a pretty significant jump in expected speed/facility from secondary expectations. Even students who don't have any processing issues can find themselves scrambling to keep up.
