This woman is newly, newly sober and is about to lose her Mother. Early sobriety is afwul hard (thirty days is great but she's barely out of the starting blocks). And if her Mom's her best friend, then she is processing a major, major life event.
I'm going to be your GF's advocate here for a moment. You have told her you'll leave if she doesn't stop - so she's stopped. For the relationship. Because it's what you need her to do/be to stay with her. That's a major life change - if she sticks with it, great - but if not then what? Being who she is right now is not enough and she has to deal with that as well as lose a parent.
You want to be with her. You know her better than anyone (after eight months). You want her to be involved with your daughter. You're amazing together.
It sounds like you've decided what this relationship "is" - and who you need her to be - before it has even really started. That's codependency.
On her part, is she maybe looking for a life preserver during a time when she's about to lose the most important person in her life? People will grab onto all manner of things when they're faced with a major loss. That's why folks who have been married decades up and remarry within months - it's not because they didn't love their spouse, it's because that hole is enormous and they're looking for someone to fill a part of it.
Good relationships start with a solid foundation. Trying to get to know someone while they're going through a major life event is hard. Even if she weren't an alcoholic, this woman might not be emotionally available in the coming months simply because of grief and loss. She's going to need every ounce of energy to deal with this - let alone work a program at the same time.
You seem to be going into this with a clear picture of what you need - I'm not sure you have a clear picture of where she's at and what the odds are that she can give you what you need as well as be there for herself right now.