Yes, it makes sense, but...You make the decisions regarding your daughter and it doesn't matter if her alcoholic mom thinks you are okay with her drinking. As you say, if she wants to drink, she will drink. Actually, from what you have written these past months, it sounds like she is past having any choice in the matter and she will probably die from drinking.
I suggest you not concern yourself over what alcoholic mom thinks and continue to do what is best for your daughter. She is the most important person in this whole situation.