First, neighbor was wrong to invite her, and you were right to let her stay. However, if she stays with grandma on the weekends, she is a neighborhood kid, so if you invited all of the neighborhood kids to the party, and excluded her, that's wrong. Honestly, she sounds like a typical 11 year old girl. The amount of exclusion that goes on up at school with the 5th and 6th grade girls is NUTS. Plus all of the instagram photos and group texts... Thank goodness we're almost done with this phase (I've found thinkgs get back to normal in high school). So her saying "you can't play with us" wouldn't phase me at all (I'd just say, yes he can). I'm assuming she's had a troubled life, if she's living with grandma on weekends. As for playing with gas, heck, when ds17 was in 6th grade, a bunch of boys had to be taken to the police station, because they were building bombs on the elementary school lawn! Boys do tend to get into more trouble than girls.
Also, you state that your dd9 likes the girl, yet she didn't want her to be invited? Could you have suggested not inviting her?
Also, you state that your dd9 likes the girl, yet she didn't want her to be invited? Could you have suggested not inviting her?