Yep, you said it, she's a petfectionist. I'm a recovering perfectionist myself. She likely hates herself and is projecting that outward. She's not happy. The cult is one of perfectionism and subsequent control.
I don't know how to make her want to be happy. I wanted to be. There are cognitive distortions with perfectionism that are easy to point out, but a person has to want to be happy more than they want to feed their ego. There has to be humility at some point.
A book that helped me was "Happier" by Tal Ben-Shahar. It is about perfectionism and control.
I think the only chance you have with her is to get her to acknowledge her unhappiness and to get her help on a human level. Once she's a healthier human other dominoes can fall. At least that was my awakening process. I was deeply unhappy and searching for happiness. I found emotional health and then I could see how toxic the cult was.