Why does Dr. King say that civil rights leaders “have no alternative” but to demonstrate in Birmingham?-

Actually King explains it best.

From the text:

Then came the opportunity last September to talk with some of the leaders of the economic community. In these negotiating sessions certain promises were made by the merchants, such as the promise to remove the humiliating racial signs from the stores. On the basis of these promises, Reverend Shuttlesworth and the leaders of the Alabama Christian Movement for Human Rights agreed to call a moratorium on any type of demonstration. As the weeks and months unfolded, we realized that we were the victims of a broken promise. The signs remained. As in so many experiences of the past, we were confronted with blasted hopes, and the dark shadow of a deep disappointment settled upon us. So we had no alternative except that of preparing for direct action, whereby we would present our very bodies as a means of laying our case before the conscience of the local and national community.