Validation, validation, validation.
Their BPD dad invalidates their experiences, so you have to double up and provide even more validation than in a typical healthy functional family.
My son was 9 when I left, and he had to visit with his dad on his own. The more in touch you are with how you feel, the more authentic and powerful your validation will be. That might mean recognizing that your son's body tenses when he talks about his dad, or recognizing that he gets quiet before he goes over to his dad's. I remember after one incident with S13 (9 at the time), I told him that adults find it very hard to be around N/BPDx -- it's challenging even for grown-ups, and that means it will be even harder for a child. I told him that sometimes he will need to make choices to get along with N/BPDx instead of being right, because that's safer. And that it's ok. This was particularly true when it came to N/BPDx saying things about me. S13 said he tried to never say my name or refer to me because N/BPDx would get mean. I think he felt guilty because in truth he was siding with N/BPDx so that he could bond with him -- I told him that I wanted him to do what he felt he could handle, that as he got older he would probably learn new skills that come with having more experience. I also tried to use interactions as a way to talk about these skills. If he had a bad interaction at school with another kid, we would talk about it. What happened, how S13 responded, what the other kid did, how he wanted to resolve it. Sometimes he made the connection directly with his dad, sometimes it went unspoken.
I had to be patient and not expect more from S13 than I was able to handle living with N/BPDx.
Our kids will encounter many difficult people in their lives. BPD parents won't be the first or last to bully them or behave abusively. We can help our develop skills that they can use everywhere, and give them a stable sense of self by validating their perceptions and feelings.