Hi Imatter!
This sounds eerily similar to something that I can see happening with one of my sisters, and I anticipate it happening very soon. This post will hopefully help me prepare a bit for how to respond. I'm sure that it was very hard to hear that she's been hurt, regardless of if it's your responsibility or not. A couple things that you wrote stood out to me.
She defends my mom passionately saying she just is a person that needs a lot of reassuring (with a small acknowledgment that this is sometimes frustrating) and full of love to give.
If your mom is anything like my mom, then this statement would be stretching the boundaries of belief quite a bit. If you and your husband are able to see the reality of your mom's BPD, then it is not something that is well-hidden. Your sister is either willfully ignoring reality, or unable to accept what the reality is, because accepting it is hard. Very hard. From this, I would say that she is fully enmeshed. She is hurt because she is feeling your mom's feelings for her. Your mom is hurting and feeling abandoned (I would assume); therefore, your sister is feeling the same. I have found myself doing this with own mom SO many times in my life. Honestly, unless she can be forthright about something that you have specifically done to hurt her, I would leave things where they are. I doubt that she can really be honest about what is really bothering her right now (e.g. family tension, etc.)
Should I just resolve myself to the fact that our relationship is not going to be deep because she is too much on mom’s side right now?
To again reference my aforementioned sister, I can at least give my own experience with this. My sister has always been the golden child with my mom (except a really weird year where she wasn't. I never could figure that one out...) Anyway, my S did finally go to counseling, and really the only thing that she told me about it was that the counselor said that she was codependent. I talked about that with my mom once, and she told me that both she and my S both that it was ridiculous. So that was her counseling. She's still very much the golden child among my three siblings, and still very enmeshed with my mom. Anyway, since I have been NC with my mom, our conversations have been more and more strained. We don't talk about any family matters through an unspoken agreement, and I have an uncontrollable urge to say every nice thing that I can think to say about my H when we talk. We mainly talk about our jobs, our health, our work out routines, and other random life stories. That's about it. Unfortunately, I think that I mentally link my sister with my mom in my mind. What I have decided is that I love my sister, but I cannot have a lot of contact with her and fully heal, even though she's not the one who I'm healing from. But, I know that in the future, I will be healed, and then that relationship can again be what it was. You might not be able to have a deep relationship with your sister right now and still do the work you need to do, but that doesn't mean forever.
Good luck!