Nope what you need to go do right now is go see your disability team.
If your university have said that you don't have to do presentations, your professor does not get to decide that doesn't apply for their course just because they're ignorant about the impact that anxiety and ASD have.
The fact that you are this upset about doing it is exactly why your professor doesn't get to decide. Your tutor, your disability contact, a student services representative, anyone that is meant to support you, you need to contact to inform them of this situation. This is not supposed to happen.
And while presenting is a useful skill, throwing yourself in the deep end is not the way to do it. It just causes more anxiety and makes it even worse. Working your way up is a good way to do it, and you've already started that by giving presentations to just the teachers. You're doing great.
Anyway, your professor is wrong, and this is almost certainly against your uni's policies. Go find someone in charge of that to get them to tell the professor for you. If you have more questions or run into roadblocks let me know.