Hi Stalwart,
Hello to you, and I'm glad you're here posting. I'm sorry for typos or clunky sentences here, I'm writing quickly as I'm about to go offline, but saw your post and wanted to respond.
My son talked about not wanting to live when he was 8. For two years I tried to manage these conversations by saying all the things a parent would say, similar to what you mention.
It was when my son was 10 that I learned about validation, specifically for people who are experiencing suicidal ideation. I always felt I had to try and lessen the pain (saying, "you're loved, we all love you, we would miss you, you have so much to live for, you have this skill and that talent and this gift". These very loving and natural responses are speaking a different language though. Validation skills instead focus on accepting and acknowledging how the person feels. They need to know that someone is listening to the pain they are feeling.
So now when my son says he doesn't want to live, I say, "You must be hurting deeply to feel that way. Did something happen today that brought up pain?" Or something like that. There are whole books on it, and we have a bunch of materials here on the site.
Validation is one of the best skills I learned here. It must be practiced and refined, and if it does not work immediately with your BPD daughter, try it with someone who may present a less steep learning curve. When you get it right, it works.
Here are the materials on the site that might help:
I hope I am not repeating what you already know. I try to slow down and read through previous posts to get a feel for what has already been suggested.
Also, validation is one of the core skills in DBT, developed by Marsha Linehan. It's also one of the skills that researchers recognized worked with patients experiencing SI.
LnL