Hi Finding happines
First of all congratulations on your engagement!

This situation with your (former) friend, also your fiancé's sister, is quite unpleasant. It sounds like both she and and her mother use things such as fear, obligation and guilt to try and control other people. Would you say that's an accurate assessment? If so, I encourage you to check out an article we have about this subject, here's a short excerpt:
... .fear, obligation or guilt ("FOG" are the transactional dynamics at play between the controller and the person being controlled. Understanding these dynamics are useful to anyone trying to extricate themselves from the controlling behavior by another person and deal with their own compulsions to do things that are uncomfortable, undesirable, burdensome, or self-sacrificing for others.
You can read the entire article here:
Fear, Obligation and Guilt (FOG): How We Allow Loved Ones To Control UsThe issues we are facing are that my fiance's parents are desperate for resolution and see myself and my fiance as the ones who need to "fix" the situation. We have continued to communicate that there is nothing we can do to "fix" anything and that my friend's issues are her own. We continue to receive threatening, passive aggressive, bullying messages from my fiance's mother telling us we need to make amends.
This is concerning. How would you generally describe your fiancé's mother's behavior and her relationship with your fiancé? Do you perhaps believe she might also exhibits certain BPD traits?
When dealing with people with (possible) BPD or BPD traits, boundaries are very important to protect your own well-being. I understand why you have been distancing yourself from the unhealthy behaviors. We have several resources here that you might find helpful for setting and enforcing/defending boundaries. Here's another excerpt:
Boundaries are how we define our values to others. A boundary is nothing more than the outer perimeters of our independent core values - it's like a fence - anything inside the boundary is consistent with our core values and anything outside the boundary is not. For example, if your independent core value is "always to be respectful of others" a boundary question might be "would abruptly walking out of the room when someone says something offensive be inside or outside of your definition of this value?" It's not always obvious as we all see things differently. As you can quickly see, with values, we have a significant responsibility to lead, educate and inform others - we must walk the walk, have effective communication and be consistent.
Here's a link to an article about boundaries:
Getting Our Values and Boundaries in OrderTake care and I encourage you to keep posting here. Welcome to bpdfamily