I don't have any advice for you, but I did want you to know I have been in the same boat as you.
My brother is in his 40s, my mother is in her 70s. I live 3000miles away. He lives in a travel trailer on her property right next to her house. He has mental health and addiction issues. He causes her so much stress, I know it will be the death of her.
My brother should be locked up, either in a rehab, a jail or a hospital, but he isn't. There is no recourse for people who don't want to help themselves unless they have proven to be a threat to others. In other words they have to harm someone physically before action can be taken. Disgusting. ( My brother who is not allowed to own firearms (for mental and legal reasons) was found in possession of one and did not go to jail for it...WTF?!) The system is beyond broken. Not just for the addicts/alcoholics/mentally ill, but for those of us who have to DEAL with them in our daily lives. Sucks.
I don't want to hurt my mother any more than she is being hurt by my brother, but I have decided the next time I hear about her being scared or involved in something she is uncomfortable with because of my brother's behaviour... I AM going to be calling for a welfare check on her and accuse my brother of elder abuse to the police. My mother is so very codependent and has been her whole life to her parents, to my dad and now my brother, there will be no changing that aspect of her personality..and I know she is upset I moved away and that I don't want a relationship with my brother... but I will not have relationships with people in active addiction (or not in real recovery) and I wont help someone else enable someone like that either. That's not good for me and it's not good for them.
I hope you manage to find a way to do what is best for your mom. If your sister manages to find recovery as a "side effect" of that then all the better, but as you know that part is up to her. You are not a bad sister for wanting what is best for your mother. Your sister is a bad daughter for what she puts your mom through. Your sister's actions are her responsibility. Every action has consequences, maybe its time she faced hers and quit making your mom's life hell.
Wishing you and your entire family a happy ending to this saga.
*hugs*