You’d be better off just starting a thread with what you want to ask, without the game playing.
You need to understand that only (1) in your case was a true option. And it’s not your mother’s fault that your arsehole of a father lied to her and painted her badly.
Sure, she might just have been a bad mother - they exist.
But she stuck around for 12 years!
If you read threads on here, you’ll see there are plenty of threads from abused women who believe that the father is “a good dad”.
Your mother didn’t know your dad was going to turn on you.
(3) is really hard, legally. And she was fighting a dirty fight, with a total arsehole - from what you say. You cannot think that these were 3 equal options for her.
You think (2) was an option? See what happened when she only wanted 50/50? Your father fucked her over. How much do you think (2) was ever really an option?
My two most likely views on your situation?
a. Your mum wasn’t that great a mum, that happens (but as she’d stuck out his abuse for 12 years of your sibling’s childhood, I’m not seeing that as certain)
b. Your mum reached the end of her tether with the abuse. She knew he’d fight, so she agreed 50:50, thinking it was the best thing. She didn’t know he’d turn his abuse on you. Hell, maybe she had people telling her he was just controlling her and the reality was that he was a man, he wouldn’t get custody. Or - he would keep it up as it was just to try to stop her going - but once gone, her kids would end with her anyway as he couldn’t be bothered. Maybe she genuinely thought both parents should have equal access (I do, where it suits the children). So having agreed 50/50 he totally fucked her over. Blame your dad for that, not her.
I recommend that you attend counselling on your own, and possibly in the future with her. I recommend that you find out more about abuse and its effects.