Hi peachybear. I am sorry to hear your boyfriend may have BPD that you think he picked up from his mother. It is possible that they are 'traits' rather than actual BPD behaviors though. Opinions seem to vary on this but I think of traits as being less automatic and ingrained than what we see in people diagnosed with BPD, though in essence, there is no difference in methodology when it comes to over coming those problems (again, IMO). Can you describe any of his behaviors that fit the diagnostic criteria?
It is possible for him to 'get better' even if he does spend his days with his mother. The thing is though, he has to want to get better and it does not sound like he is interested or at least he is not driven to get help and change. I can't say your relationship is doomed to failure though I will say that having strong boundaries are a must and even with the best of boundaries things are likely to be challenging (at least from his mother).
Peachybear, you are welcome to post on this board but I wonder if you might find more help at Staying: Improving a relationship board. Here is the link:
https://bpdfamily.com/message_board/index.php?board=6.0I wish you good luck and I hope to see you around the boards!