AITA for trying to get my sister kicked out?

My house consists of my Mom, my disabled sister, me (20) and my sister (24) with her 1 year old son. I'm the only one working right now, and I'm trying to save up to move out with my fiancee, but after my base rent of 200 a month plus all the extra I give to help cover bills, I basically have a coin jar of savings. My fiancee can't move in with us because we have dogs and she has 2 cats, but my goal is to live with her by any means necessary.

On to my sister. She would sneak her baby daddy (abusive asshole) into the house after Mom told him he wasn't welcome, up until the baby was 6 months old. Hasn't paid rent since she last worked, when she was 3 months pregnant. Doesn't do chores, half the time she won't even clean up after her own kid.

Forward to today. I went to talk to my Mom to find her in tears. Her bank card was missing. Then she goes on to her account to find 60$ missing, from a gas station, drug store, and a liquor store. My sister stole the bank card and lied when asked about it. Since this is the dad's weekend, she's spending the weekend over at a guy's house with the card. Hasn't answered Mom's calls since she initially asked about it.

Mom is heartbroken, but I am furious. My Mom's the type of woman that would give her last penny to someone who needs it more than her, and to see this happen to her after everything she's done for my sister, living rent free to buying her son new clothes just yesterday, it sickens me. 60$ may not sound like much to some, but to this family it's a massive blow.

If she reports it as stolen, the bank will review cameras and my sister will get prosecuted. This would leave custody of her son with her father, which we don't want. I again told her to kick my sister out, and she once again didn't want to do that to her grandson.

So, I gave her an offer. My fiancee is wonderful with kids, and the dogs never go in the basement, making it the perfect place for us to keep the cats. I told her if she sued for custody of my nephew and kicked my sister out, I would give her my entire savings as well as the rest of my paycheck each month so she can truly catch up on bills. My fiancee would also pledge her disability each month to help, watch the kid, and do chores.

My goal here isn't to take my sisters kid. I don't want to do that, I know she loves her son, but she should love her son enough not to fuck over the woman that puts a roof over his head and food in his stomach. She could take her son at any point she felt ready to take care of him, whether staying with a friend, a boyfriend, or getting a job and her shit together. But I truly feel like my Mom wants to just let this go because she has a kid, and I hate to see it happen. My opinion? Unless we take drastic measures, she won't learn a damn thing.


EDIT: I had to edit this for 3k limit, but since I realized the wording was weird: I initially told Mom to kick my sister out when she was sneaking her boyfriend in, but she didn't want to do that when she was pregnant.