That is hard. If the manager is likely to side with her, then that limits your options considerably. In that case, if you truly need this job, then put your head down, persist, and let this delightful woman be wrong.
"Sure." And then don't do it. Over and over.
I'm sorry that you're dealing with this. It's not you; it's them. I suspect you're wonderful and I'm sorry that your employer doesn't recognize that.