MIL with borderline personality disorder

Fact is this: she has a mental health issue that poses serious risks to you, your DH, and your child that are going untreated because she doesn't feel like she needs treatment.

The fact that she has no interest in trying to learn to cope with, or learn strategies to, mitigate her emotional and mental issues is a red flag.

Your boyfriend, well, it's his life. If he wants to subject himself to her untreated BPD that is his choice, but it doesn't have to be yours and it certainly shouldn't be your child's.

Fact of the matter is having a healthy is not a free pass to act shitty. It's understandable why your MIL acts shit, and I empathize with her, but that doesn't mean I want to be around her.

Her choices affect you if you are around her. It's not unreasonable to say:

I don't want to be around your Mother and I don't want her to be our child because she is making no effort to treat her very serious condition. By definition Borderline personality disorder is a mental illness marked by an ongoing pattern of varying moods, self-image, and behavior. These symptoms often result in impulsive actions and problems in relationships. People with borderline personality disorder may experience intense episodes of anger, depression, and anxiety that can last from a few hours to days. I don't want to sacrifice my mental or emotional health to be around someone who isn't even trying; and I don't want her to be around our child because her outbursts, and wild mood swings, are by their very nature makes her an unstable person and children need stability more than anything else. It's not her fault she has BPD, but it is 100% her fault that she is choosing to do nothing about it.

I mean it just is what is.

If your boyfriend wants to sacrifice himself to placate someone who is making zero effort that is his call but he shouldn't expect the two of you to do the same thing.