If you go to the raised by narcasists subreddit you will find that barging in on teenage children using the bathroom is a very very common thing that narcasistic moms do. It's very strange that all narcasists parents seem to play out of the same playbook. You should take a little bit of money and get a lock installed on that door.
In a lot of your language in your comments you talk about caring for your and appeasing your mom as if she's a child - the story about watching TV comes to mind. But then you judge your young siblings through a harsh lense and expect them to behave like super mature adults in a crazy situation.
I actually kind of think you should be proud of your sister for standing up for herself and requiring more from your mom. And it's actually sweet that your other siblings support your sister and take her side and that there is unity. You seem to want to keep the peace but in effect what you're actually doing is indirectly taking your moms side and making your sister feel like her concerns and upsets don't have value.
I relate alot to your sister and had an older sister very much like you. It was crazy making to know I was right and that mom had issues but was always being the one told to appease and manage mom and control my temper. I was the child in the situation but I would be delt with harshly and my mom would be the one to be coddled. I thought my elder siblings were crazy and I still do. In the end I moved away as soon as I turned 18 and I hardly talk to them anymore.
It's funny, my family call me argumentative and difficult. But to everyone outside the family I'm known as the biggest doormat and a super passive and quiet person. The reality is I'm not argumentative! I just never allowed my family to walk all over me and dictate my life. Your sister is a strong one. Despite the madness she has her voice and stands up for herself even at a young age. That should be encouraged not silenced and steamrolled.
Sorry if I've left too many comments. Perhaps I feel attached to your story because I relate so much. I'm going to reocmmend you this club I'm a part of called homegirls United. Its a charity for elder daughters of immigrant families. They're a friendly bunch and often get together for events. I think you'll find your people here.