OP do you have an admin that can manage your Twitch chat while you stream? They can look out for her handle and deal with her when she gets rude. Sometimes having at least one person who will call out bullish is what you need. I don't know if Facebook has a similar function but I would do it there as well.

Honestly, I know you don't want to deal with the fallout, but she needs a hard lesson in what happens when she crosses boundaries. My brother was a dick to me on social media too; talking doesn't work, blocking him was the only way to get it to stop. Was he pissed and threw a hissy fit? Yeah, then I ignored that. If you and your sister aren't living together what could she do that makes it not worth blocking her?