We have learned a lot about human cognition since the 1960s that would have served MLK better in his approach. When people make decisions, big or small, their first impression becomes anchored and is tough to overturn even in the face of strong evidence. And if that impression is stated publicly the effect is even stronger. If the decision maker acts on the decision, it is stronger still.
So MLK should have thought of a way to get the incoming Birmingham administration to do something publicly, no matter how small, in support of his movement. It didn't even have to be directly relevant to equal rights. That could have come later. Instead, he forced their first act to be directly opposed and guaranteed that they would continue to oppose him.
Perhaps he was more interested in gaining national attention and preferred a public conflict. That is often what civil disobedience is designed for. But not for influencing the local pols, he did the exact opposite of what might have worked. Of course, MLK did not have the benefit of the past 40 years of cognition research.