UPDATE - UPDATE - UPDATE
I tried to take the high road and explain to her why it is important to invite the fat girl... treat everybody how you would want to be treated, etc. She sulked for a little bit, but at the threat of not having the party at all, she wrote the invitation.
She came home from school, and I found the invitation still in her backpack though. So I told her if she came home with it again on the last day of school, there was going to be a big problem.
Well she didn't come home with it, but we got a phone call from one of the boys mothers asking if her son was really invited to the party... she had crossed out fat girls name and given it to the boy in her class that she likes.
That was the last day of school, and I have no idea where fat girl lives or her phone number, so my daughter managed to stall the process long enough that she didn't have to invite her. In the end, she did give me a rather compelling argument (for a 3rd grader) that everybody would just be making fun of fat girl and it wouldn't be any fun for her anyways. It's a good point. I am rather glad to not have to worry about constantly policing the party for bullying.
So now the party is going to be 10 nine year old girls and 1 nine year old boy. The girls are spending the night. I'm kicking the boy out at 8pm. To the credit of the boy, he's definitely the little ladies man, and a perfect gentleman of a kid. I have no problem with him coming. He comes over for dinner, cleans his plate, asks politely for seconds, and then clears his plate all the way to the dishwasher. He washes his hands after he pees, and says please and thank you like he doesn't know how not to. A+ kid.