IHWIMY, I am sure there is more going on than what you described in your original post. However based on what you described in your posts, it doesn't sound like bullying to me, it sounds like the other girl did not want to be your daughters friend. That is legit.
I have WHAT in my yard? :
And PM?
But trying to force someone to be your friend just doesn't work. And that sounds like maybe what DD has been doing. For a month? it's been clear she wasn't welcome in that little circle and yet she keeps on walking into it. While those snotty ones absolutely don't have to be jerks about it, they also shouldn't be forced to hang out with someone they, for whatever reason, don't like. They'll have to deal with that when older, lab partners in school, same school team, later in their job. That's work stuff... teaching them now that they HAVE to include someone in their personal lives that they don't want to isn't a good idea.
This is mean, and doesn't apply to kids this age, or a classroom. They are all forced to hang out with each other in a classroom all day no matter what. It is just like work and to pretend it is not is not helpful. Snotty ones need to learn to cope with having to deal with people whose social skills (or social status) may not be up to their standards.
I agree with PM that forcing someone to be your friend won't work. I also think it is a very bad thing to tell children that they can't choose who to be friends with. How would you feel if someone made you be friends with everyone you were friends with for a few months in fifth grade? How would you feel if you were forced to have a relationship with every guy you've ever dated? Even the really awful ones? Doesn't that validate stalking? Friendship is a privilege not a right.
Every story has two sides, have you thought about how odd some of your daughters behaviors may seem to the other girl? Do you know what the other girl might be telling her parents? Maybe she is being a "mean girl", but maybe there is another side to the story.
Now a confession: I was a very awkward kid! I had a hard time making friends, elementary school was very difficult so me. I had to learn how earn friends, somehow it became a lot easier around high school. Some kids just aren't very good with a certain time, but outgrow the awkwardness.
You know your daughter is lagging in her social skills, and helping her is the only thing you can do. You can not change the other kids behavior and I wouldn't dwell on it too much with your daughter. Focus on the things that you can do something about. I agree with rodriguezpoultry that getting a dog or changing schools might help.
Good Luck!