I have also had (have) a friend, "Laura" (not her real name), who is an alcoholic, which I'm afraid to say is what your friend is. I have known Laura for 10 years and she has been sober now for 5 years after going to rehab (fell off the wagon the 1st time and had to go back a few weeks later). I too was concerned about confronting her about her behaviour in case of the repercussions but decided that in the end I didn't care if she hated my guts for the rest of her life, if it prompted her to get the help she needed then it was worth it.
We are both members of the same playgroup and with hindsight we all enabled her behaviour by allowing her to carry on the way she did and not calling her on it or confronting her. Her problem became obvious to various members of the group gradually during the first year and by the 1st anniversary of our group forming there was never a time we saw her sober (10am, 1pm, 11pm, it never mattered). The main reason none of us said anything was she had a child and we were worried that if she cut off contact with us we wouldn't be able to keep an eye on the child. (We found out later that she had been drinking continually all through her pregnancy but this was before we knew her. Had I known her during that time I would never have befriended her).
Fortunately she never became violent or nasty, she just rambled, got weepy, or was a bit out of it. Looking back now (isn't hindsight a wonderful thing) I would have confronted her about it a lot sooner. It wasn't until she went into rehab that I realised what a physical and emotional drain it took it on my body to be worrying about her all the time. This might sound hard but if she took up drinking again, or I met someone else who was going down the same path, I would cut them off right there and then. Once bitten twice shy.
If I was in the same position again, either with her or someone else, I would definitely not invite them to the party and I would tell her why when she found out about it. If your other firends don't want her there either then that's telling you that they are all sick of her behaviour as well, and I would hate to see you missing out on a nice celebration for your birthday if they chose to boycott if your other friend is invited (is this a possibility?).
Sometimes a line has to be drawn somewhere, and a wake up call given and this might be it for your friend. But don't be surprised if she refuses to admit there is a problem or take responsibility for her behaviour/actions. Laura did say to me when I confronted her that if I had said anything a year earlier she wouldn't have listened. But at least it might give her something to think about.