Wow, yes it does sound just like her...maybe she shouldn't be in that job position if that is her reaction. A secretary is one of the most important ways to communicate from school to parent, with an attitude like that, maybe it's time to find a new job.
As a teacher, I have encountered many parents that think they should be running the inner workings of a school. Know your place. This secretary is a staff member of the school. This is her work, not yours. If you want to help out at the school, that is great. Running to the principal is another matter. I can promise you that the staff is talking about you trying to get members in trouble and they are laughing behind your back. Know your place.
The best advise I can offer (and use myself) just keep everything EVERYTHING she says does etc, documented at home. At our charter school, we had the same issue but throw on top the fact she was throwing her religious beliefs at us and we had a mess!
Just keep your head high, you know what your intentions are and the best that you can do at the end of the day is look at your own relection knowing you did what you can do.
Our secretary crossed a line, took a few years, but she did. What did her in? The PTO purchased a new US and Colorado flag and she announced that was a bad purchase as she would make sure they would never be hung in the school. EVER. 8 years later, they are STILL standing proud in our new auditorum. She isn't.
She'll slit her own throat with this behavior and with a personality such as hers, there is not much you can do to get around it other than stroke her ego. That said, the environment of any school is affected by the person at the top, in this case the principal. Seems to me he is the real problem here. Presumably there are other parents who are not happy with how things are being handled in your school. Maybe you need to band together with other parents to try and get him out - that's the beauty of a private school!
Typical sociopathic behavior. I knew a woman like this who had everyone tied into knots with her freak, look-at-me-look-at-me, hot/cold behavior. The manipulation was so disconcerting and puzzling. Eventually, she took it one step too far and her house of cards fell in around her. Not too long afterwards she told another person in our circle that she felt she'd "burned her bridges" (no kidding), so she moved away. That was a happy day.
In your situation, since the success of a project is so dependent on this "loose screw", frankly, I would finish the project to the best of my ability and wouldn't volunteer for a repeat. I just don't have the personality that could handle having to step into the ring with someone like that again. Instead, I would help at the school in some way that didn't involve her.