what’s the best strategy for getting someone who’s in denial about anything being wrong with their behavior to recognize that they have BPD and seek directed treatment so that we can all work together towards healing as a family?
My short answer is you probably can't.

Sounds like a great goal, and a win-win for everyone. But if that worked for BPD, this site wouldn't have so many members.
Trying may escalate the problem, which it sounds like you already know.
Great response from NC already. To that I would add that what has worked for me is three things:
- educating myself about BPD, and learning new relationship skills to navigate around the intense emotional roller coaster "feelings" (fear, anger, rage etc) of my uBPD mom. (To be fair sometimes her feelings could be intensely positive too, but this is usually followed by a crash where she would dump her feelings on me)
- accepting that the only thing I could change (that I am in control of) is how I react to her.
- letting her make her own choices (even when I think they are bad ones), and learning herself from the consequences of her own choices. This lets her have complete control, and means she doesn't feel invalidated by my disagreeing with her choices. I
may ask her validating questions, but I
never give advice any more.
I use SET,
no JADEing, validating questions, boundaries, radical acceptance, and I'm now a believer in mindfulness and self-care. I also have less contact than I've ever had before (about 2X/week) , and it's currently working. She's 84, very frail, and chooses to live independently rather than go into assisted living which she qualifies for.
I want her to get the help she needs and recognize that she exhibits behaviors consistent with BPD but this is hard because she’s so quick to point the finger at everything everyone else needs to change.
I can completely relate to the cycling you describe. My mom use to RAGE at me. In my experience, if you try to rationalize and explain to pwBPD, it will go very badly, ie if you try to change her

... well this might be a non-starter, and do more damage. If you try to change yourself, it will go a lot better.
