To the folks who are saying the troop leader is wrong to have a mother-daughter event, it should be changed or not done at all, etc.: Really? Children as they grow up will encounter many such situations. Should they learn to handle them (painful as it can be) or should things be tailored or shut down to accommodate every possible family situation? So no dad-and-daughter events either, no grandparent events, nothing. I lost my dad when I was 12; nobody revamped the dad-and-daughter events at Scouts or school on my behalf and I would not (even at 12) have expected them to. The event, by the way, may not even belong to this one troop; our service unit (grouping of a dozen or so troops) does a mother-daughter dance and a father-daughter dance each year open to girls in all those troops. So the troop may be participating in a much larger event. I notice that in her post, M. herself did not ever call for the troop not to do the event at all, but some people posting feel it's fine to blast the troop. I don't see how changing or canceling events will help anyone learn.
And the girl does not have to attend. This may be the year, M., that she should sit it out if she feels like this. How far away is the event? She may change her mind close to the time of the event. I think the best you can do on that front is to tell her, "I know it's painful and I know that some kids have been teasing you about my being there instead of your mom. It's OK if you don't want to go this year. If you change your mind, I'm glad to go with you but this event is optional and is meant to be fun, but if it is not fun, of course you do not have to go. If you want to do something else that night, I'm glad to take you out somewhere for a movie or anything, but you can let me know later." And leave it alone. If she seems to warm up to you closer to then but it's too late to register, for instance, you could invite her out to a girls' night doing whatever she likes best (movie, dinner at a special place, etc.).
I'm glad the troop leader will address the teasing. I am sure that had the leader heard it for herself she would have nipped it then and there, so it is good that you let her know about it happening; it is NOT how Girl Scouts should ever act. As for teasing at school, please tell both her classroom teacher and the school counselor immediately -- and your brother should relay the message along with you.I would see if the counselor would meet with your daughter privately about it and maybe get her to open up. She is clearly in a phase of processing the fact she's a preteen and is entering those difficult years without her mother there. She is resenting you for not being her mom--which you can't help. She is too young to appreciate just yet what you're doing for her, but stick by her even when she says she doesn't want to see you and wishes you weren't her aunt. You are doing amazing things for her even when she does not realize it.
I would continue to pick her up if you are part of the routine that helps her dad out. Don't give in there; just be very matter-of-fact and have her dad, not you, say "I know you said you don't want Aunt M. picking you up. But it is important because it helps me (whatever) at that time. So she's going to continue to do it and I hope you understand how she is helping us all." And not a lot else, no lecture about how wonderful you are (though you ARE) because the girl will just get defensive about her own feelings and will resent you more. She will come around but it could take time and a lot of patience from you. But I bet you have it already.
Talk to your brother about any other signs he is seeing. Is she moody and withdrawn? Is her schoolwork suffering and are grades slipping? Is she no longer interested in friends etc.? If those are going on -- he should get her back into counseling where she feels safe venting about her feelings and her mom.