Accommodations cannot pose a financial or administrative burden, or substantially change the academic program. These things are slowly being defined in court but the language is vague and open to interpretation.
However, it would seem right now that the problem isn’t that the university is saying no, it is that they are avoiding it because the administrator in question doesn’t know and hasn’t gotten around to finding out or doesn’t know how to find out. This is why legal advice can help everyone, because once info is known, a resolution one way or the other will happen. It may be that no one on campus (or even at the state level) knows and it will take them work to figure it out and they are avoiding it.
Be aware that they are always concerned with setting precedents as well.
I would work on this with an advocate, while working further with a psychiatrist, possibly trying beta blockers. You just take one before the speech, much as musicians do before a performance.