What if we just say , we don't think you are mentally ill, but
we want you to go to talking therapy to just help with interpersonal communication skills.
Another possibility is to let her know you are taking a time out to learn how to keep yourself safe from these emotional outbursts.
She may not have the capacity to fully reflect on her actions because her pain is so intense and her behaviors are so desperate. Pointing out yet another defect in her, and the need for her to get help, will likely feel like adding insult to injury, even though her behaviors are the ones that are interpersonally unacceptable.
It's really about being effective. Giving people ultimatums usually doesn't work, especially for people who struggle to control their emotions and feel such profound confusion about what's happening, and why.
My SD was an intense walking ball of need unlike nothing I had experienced before. She was more covertly aggressive than overt, probably more of quiet BPD type than externalizing. Trying to reason with her backfired, and trying to get my husband to control her backfired even more.
The only thing that worked was focusing on me and letting go the illusion that I had control over what she did.
Once you feel ready to be around her, once you've learned some of the skills that can help in these difficult relationships, start small. Maybe just you and your H with her. See how that goes. Sort of a love and logic approach. Short visits in public if that helps. Give yourself an exit plan so you can up and leave to protect yourself in case things don't go well.
With my SD, she goes to therapy but we don't think she's honest with her therapist. I'm sure if we did family counseling and shared our experiences and observations she would get a BPD diagnosis, but that isn't ever going to happen.
When I realized she isn't going to get a diagnosis and won't get the kind of therapy she would benefit from, I tried to learn what I could about DBT and will mention things as applicable. I'm not sure how helpful it is for her but when she is in crisis mode, I find it helpful for me to say, "Have you tried boxed breathing?" instead of getting pulled into solving a self-created crisis she is fully capable of solving herself.