P.-
First -- so often, we avoid difficult situations (esp. with our kids!) because we don't want to hurt them -- and in the long run this hurts them more. Talking to your daughter about her behavior would be hard, but hearing from another girl that she's not being invited...that's definitely worse.
It's important to remember that there are two systems at play: your daughter's behavior, and your daughter's self. Her self is perfectly fine. It's the behavior that needs adjusted.
My son is very sensitive also, but like every human on the planet, has things on which he could improve. I've found that by separating his behavior from his self, it allows me to get my message through without passing along shame or blaming.
My daughter is turning 7, and she has a tendency to be bossy; she just wants to play her way. We've been working with her on it, though, in a few ways. First, by focusing on making sure she understands that it's a behavior issue and not a 'her' issue. Then, when she has difficulties with someone (usually her brother) over being bossy, we talk about what went wrong and why, and how it could be done differently. And if it's blatant, she gets some sort of discipline (sentences, time out, etc.).
When I play with her, she likes to tell me what she wants my Barbie to do or say. I try to then model more appropriate behavior by saying something like: OK, I'll do that this time, but then next time I get to do it my way. OR, ok, I'll do that, but then your girl has to say or do this.
Kids often don't know any other way, and don't have the life experience to conjure up alternatives. They don't see others and their behaviors the way adults do. To give her concrete examples of alternate ways of dealing with situations could really help. And also sitting with her and being honest about specific times her behavior has created problems, and helping her see how it caused problems, and then suggesting alternate strategies for next time could be extremely helpful.
The hardest part is done for her: her friend hurt her feelings. You can help her navigate through it all, though, and help her understand what it is she does that people view as difficult to deal with, and give her ways to handle things that will make her company easier to swallow for other kids.
I hope this helps you; good luck to you both.
***Having read some of the other responses, I'd like to add:
Leadership and bossiness are not related: one is assertive, the other aggressive. You can take aggression and train it into assertiveness, but you can do so with passivity as well.
If her bossiness has reached the point that it is causing rifts with her friends, then there is an issue that needs to be dealt with.
People generally like confident individuals who are able to make decisions. People dislike tyrant-types who make decisions without regard for others and their wishes. It's a fine line that even adults have a hard time navigating, but you can help your daughter find her balance, which, I agree, is key. In finding balance, and appropriate times to take charge, and cooperative ways to lead, you can help her learn strategies to be an effective leader as opposed to a bossy style.