I've read all your updates. I personally think you should let your older one go without the younger one. I understand that the younger one will feel bad that she's excluded, but how will the older one feel if she can't go at all because the younger one isn't coming? Either way, there will be upset feelings. At least with the older one going, it wasn't your doing. It's up to your friends to invite whomever they wish.
I get that it is problematic with *this* particular family, but does that mean that your oldest will never get to do something by herself with them? From their perspective, I can understand why the mother would want to do something for just her middle child. I imagine that it's hard to carve out special moments for just one kid and especially the middle one who happens to also not be the only boy.
We actually have a very similar situation tonight at our house, although the kids involved are much younger and the situation is reversed. Our friends have two boys that are 5 and 3, and they have been friends with my children all their lives. The older one is just three months older than my son, and he will be coming over to have a sleepover with my son tonight. The younger one will stay home, primarily because his mother doesn't think he can handle an overnight without her. The younger one is very unhappy he can't come. My friend said that they are going to do something special tonight for just him because he won't be able to join his brother.
From my perspective, I have to say that, although I really wouldn't have minded the younger one coming, it's kind of nice for my son to be able to play one on one with the older one. The two brothers are so often a package deal. The dynamic of 3 kids instead of 2 can sometimes be difficult, because someone is always being picked on. As the host parent, it will be a lot easier to manage just the older boys.
It's sad for your littler girl, but I'd say just try to do something special with her. Focus on what she gets by staying home (alone time with you) rather than what she is missing.