Sadly, I think that most of us here have some experience with what you are talking about.
I agree with almost everything that
ScottishKin said. I'm not sure that I would start a discussion about why you felt the need to lie though. That might just make your situation worse.
People who present BPD traits typically experience emotions in a far more intense way than others. This means that something that would only mildly upset most people can be felt as catastrophic to others. Fear of rejection and abandonment are at the top of that list for pwBPD.
If you bring up why you felt compelled to lie, that might make her think that you are blaming her for your choices and actions. If that's the case, intense anger may be what follows.
Another possibility is that she'll agree with why you felt that you needed to do it, shame will be her response, that will trigger the fear of abandonment and intense anger may be what follows.
It is probably her reality that trust has been broken for her. If she does have BPD, she may very likely be thinking that what is going to follow is your abandoning her. All of this is probably very real in her mind and her emotions will have more effect on her than your words.
You're probably going to have to slowly rebuild her trust. It is going to take time and consistency. Those two things are essential to helping a pwBPD feel safe.