I read Washington's autobiography Up From Slavery * some years ago, but would like to seek clarification / quibble with your question.

Segregated but equal - my impression was not that this was his ultimate goal. Rather, it was a practical, short-term recognition of 'it is what it is,' combined with brilliant political skills in dealing with the dominant culture.

For his time, his immediate goal for the African American community was (1) don't be an object of charity, (2) or murdered by a white mob. In the medium term, the vision was something roughly akin to the segregated ethnic cantons of the Northeast - an Irish section, a Chinese section, a Jewish section - all somewhat segregated, but all with their own economic and political power and established, skilled white collar and artisan classes. He spent no political capital or rhetoric in pie-in-the-sky idealistic talk beyond goals achievable in his lifetime.

*His autobiography is public domain and a good, quick read, definitely worth your next uninterrupted afternoon.

I would recommend checking out W. E. B. Du Bois' work "The Souls of Black Folk" which takes on this topic, along with other stances taken by Booker T. Washington.