My last word, since I can see where this is going. Bullying is ganging up on someone in a weaker position. If this girl is excluded at 7 years old, it is not she who is the bully.
I would advise her mom to "take it up he chain" until she got satisfaction.
I disagree with your definition. According to the dictionary:
Bullying - Use superior strength or influence to intimidate (someone), typically to force him or her to do what one wants.
Bullying consists of three basic types of abuse – emotional, verbal, and physical. It typically involves subtle methods of coercion such as intimidation.
Excluding someone who has proven themselves a bully is not bullying...as long as the girls treat her no differently and do not taunt her about excluding her. In this case, I would guess the girls in the troop have no idea this debate is going on. So, they will not have any reason to treat her any differently if she is not invited to join.
The girls in my troop have no idea how many girls I have turned down (sometimes to keep my numbers low and other times because I know the girl would not be a good fit.).
I know one girl who at age 6 attempted to drown another kid in the pool to get possession of a pool toy. Her Mother witnessed the whole incident (I was the one who forced her to release the child). Her Mother's response? "She really wanted that toy and if the kid didn't want it taken, she shouldn't have brought it to the pool!" At 10, this girl is the biggest girl in the class and verbally torments the smallest girl. I have witnessed her whispering horrible things into other kids ears. Her mother has witnessed this too but always claims that she doesn't mean it "that way" and we just don't understand her and we are "bullying" her child.
I have denied this child entry to my troop twice and I don't regret it at all. It is not my job to "fix" someone else's child. While she might benefit from a positive role model, without parental support, I doubt my every other week influence for an hour would help. I probably would have quit myself due to the frustration of dealing with both the child and the Mother.