I need more information. Do your black friends belong to this organization? Were they the people that invited you? If yes, what did they say when you were rejected? Did you know anyone in this organization?
You say, this is a "mostly" black organization, meaning it is not a black only organization. Why were you excluded and not the other people?
What is this organization's purpose - and please be specific? How did you find this organization? What did you think you had to offer? What were you willing to help with?
You say a few people were determined to keep you out and called their reasons phony, but don't tell us what those reasons were. Please tell us what those reasons were.
Were you completely quiet and never said a word or did you speak up? If so, what did you say? If the meetings were organized, did you speak out of turn?
I have no idea how you define friendship. In regards to your black friends, do you go to dinner at each other's homes, babysit each other's children, go on vacations together, visit each other in hospital, help each other with aging parents, help each other fix things around the house, etc.? Do you know the names of your black friend's children and do they know the names of your children? Most important of all, do you confide in each other about personal or private things? Do you ask their advice, or seek their support or comfort when you are having relationship, work or personal problems?
You say, you shared this experience with a "few" close friends, which suggests these were white people. If so, that would explain the reason, they thought it was "reverse racism".
I suspect your answers to these questions would explain the reasons you were rejected. I also suspect that is the reason you did not tell us what actually happened.
It is easier to characterize strangers, than it is to reflect on your own behaviour.
You say, you "don't intend to just take it" - before you act, I suggest you think it through. This may be hard for you to understand, but your desire to let us know their rejection did not hurt you, in any way, suggests you went to this organization believing they needed you, and would be so grateful for your help, they would make you a spokesperson, or put you in some elevated position. Is that true? Were you treating people like a charity case?