Hello t and s, ,
We have a s6 and his father has BPD and Paranoid Schizophrenia, and whilst he had been largely unaffected by his fathers illness, I too noticed a change in his behaviour this year.
When my husband started to seriously unravel he was 5, so I knew I had to act. I involved the school straight away making them aware that his father is mentally ill. They allocated a named support person for him to speak to whenever he wants. As of this new term I asked if they could formalise it and they are paying for a play therapist whom he sees every Monday. Our son has been involved in this and is delighted that he has someone for him. He knows I see a T and it's on the same day as him.

he is fully aware that his father has a P etc.
He is also linked in to a local Young Carers Play Group, and whilst he does not care for his father he has to caretake him emotionally. At this group he mixes with other children who have parents with mental health needs.
I have been honest and open about his fathers illness with him, keeping it simple and child led is key.
Having a mother with BPD most be so so difficult and your son is letting you know he needs some help. Start with your family doctor and get him to refer your son for an assessment, the school can do the same. ( I am in the UK )
The other thing is that your son is too young emotionally to fully understand the techniques we use for our SO. Children act out behaviourally because they cannot verbalise the emotion, your sons emotional cues and triggers will be skewed, you already know this but it does not mean he has BPD.
I cannot imagine how you must feel it must be awful, but your son is young so act quickly - a play therapist will work really well for his age.
Sorry if the response is a bit wordy I was rushing whilst son eats breakfast.
Take really good care of yourself and remember there is lots of help available for your son.