I have tried to get my son into counseling several times. At first I wouldn't bring up the BPD diagnosis of my ex-wife. Now, though, it's something I always mention. My ex typically rages at therapists on the first meeting though so I prefer to warn them of what is coming.
I struggle with what to tell my child and haven't a good answer for that one yet.
This is a tough one. For 3 years I didn't say anything to my son -- although I did tell him when he was 9 that his dad's behavior was from alcohol. I talked to a child psychologist about whether to tell my son, and she said, "Kids know. They may not know what it's called, but they know something is going on. You're just giving them a name for it."
My T helped me tell S13 the truth -- kids don't need labels is what she said. But she felt he needed to know that I felt there was something wrong, and that there were other adults who agreed.
When we finally had that conversation, it was cathartic for S13.
But it's really important you can be present for your kid, and not have some skin in the game when you have that conversation. That's me paraphrasing my T