Dealing with theater people who don’t “get it” about showing up on time, every time, can be really frustrating. I’ve been in that situation, and sometimes you just can’t change people’s attitudes, the best you can do is lead by example as far as professionalism. The kindest thing I can think of about your PA is that she was truly horrified by being so late, and was beating herself up already, got defensive when you called her on it (not that you didn’t have a right to be upset!), and is already wishing she’d handled the whole thing differently.
Since the producer rolled her eyes when you discussed this, this may be an issue they’ve had with her before, but it’s worth clarifying that, and asking what steps can be taken if this is an ongoing issue. Every theater has a different hierarchy, so I can’t speak to whether you were or were not considered her supervisor. It seems like that is something you, the PSM, crew manager, and producer might want to discuss as well, plus talking about who should deal with similar issues in the future. If you were the third person that night to reprimand her, I can see how she might be less reasonable by the time you talked with her. Did this conversation happen during the show, or after curtain? If it was backstage during the show, she might have been trying to focus on the show by then, and caught off guard when you brought up the lateness then.
One alternate response to her when she got defensive might have been to go in another direction, and tell her that her lateness effects the entire team, since her duties need to be redistributed. I know it’s easier said than done, but try to stay cool and not take her bait by getting into argument while everyone is still upset by the lateness, it just won’t be productive if your goal is to get her to regret and not repeat her behavior. Approaching her at first with concern, as you would an actor who is late, and then in a way that tells her she’s a valuable team member, and you need her to do her part each night, just might get her to be more conscientious in the future.