I guess you don't want that kind of sister in law in the future ,so break up.
Even though she'll win the game by making you break up with your bf, it isn't fair for your bf to not try to do anything about this situation.
If he really tried to make his sister like you, he should have talked to her and should have TRIED to get her to talk to you.
I know it would be hard ,but he needs to be on your side if you want to fight her ,but shince he won't help, just break up so you won't meet her ever again.
She got some problems, tweeting about you? seriously ? that is crossing the line even more.
She is asking you to break up with him ,so it is her fault and your unconsiderate bf's fault for your relationship so, break up unless your bf is tries to change his sister.
personally, I would not want to be part of their family at all.
the mother only cares about her daughter and not about her disrespectful behavior and rude sister.
also, unconsidrate bf who throws his gf's cry for help to his own mom instead of doing it himslef.