HappyAdult, your post made me cry. I'm certain that my sister felt the same way about my DS when I was still with AXH.
As a mom who has been single-parenting-it for longer than I've been away from AXH, I know how hard it is to cover daycare costs and I don't know what I would have done if my parents hadn't been stable, loving, and able to take care of DS while I worked. But your sister's defensive statement
"what am supposed to do? Have no life of my own? Just sit here with my son 24/7?"
made me a bit angry. My cousin and her husband say nearly the same thing: they can't afford daycare for their kids, routinely scrounge money from friends and family for "food for the kids" and yet, they have money for (a lot of) alcohol every weekend, buy season tickets to baseball games.... (They "let" my uncle pay them rent and watch the kids for free. He doesn't even get a room, just to sleep on the couch. But his acceptance of that situation is a different rant.)
I'm not assuming that's what your sister does, I'm just noting that her comment rings kind of the same as their comments. FWIW, my comment back to my cousin before I stopped trying to talk to her about it was, "Yes, how you live your life should have changed when you had your first daughter. Your kids and taking care of them should be priority #1."
It sounds like your sister's situation is a bit different. Working as a waitress, it might be possible that she qualifies for daycare assistance. Typically it won't cover all of the daycare costs, but it would definitely help reduce the cost. There are also resources available for low-income families for rental/housing assistance.
As a DV survivor, she may still be dealing with the low self-esteem and other issues that come along with leaving a violent relationship. I don't think DV shelters / advocates put a time limit on when an abused woman can seek assistance for her and her family. I think I first contacted the local shelter almost a year after I left my abusive AXH. They offer more than just the immediate shelter after leaving an abusive home; they can help her find the resources for counselling, daycare and housing assistance. They offer support groups that will help her realize she's not alone in dealing with this.
(I also believe that a young child being left in the care of some one who is routinely drunk and passes out while caring for that child could qualify as neglect.)
I hope your sister is open to seeking assistance. Wishing you and her both continued strength and peace.