You need to tell her that - admit you handled it poorly, & you're sorry, but it's because you want the best outcome for her. Regain some trust.
Then point out how so many people suspect it already, and that getting treatment for it will help hide the issue so she won't have the stigma from outsiders who don't care so much about her giving her grief. Nobody has to know except the family who already knows. Anybody else can be told her mannerisms before getting treatment were "just a phase" that she's now "grown out of."
Basically, put yourself in her shoes, and try to come up with approaches more compatible with how she sees things. Make sure you convince her that you're thinking like her, understand where she's coming from, and truly want to help her, not control her. Help her deal with this in a way that ensures minimal potential embarrassment.