Lyndon Johnson vs. George C. Wallace was a mismatch at least when it came to persuasion. Because Governor Wallace was a master at propaganda and touting his own, well excuse my French, but bullshit. Even though I doubt he believed his own bullshit, but instead use it to hold power. But LBJ wasn't someone you bullshitted without any response and LBJ would cut people off if they knew they were simply wrong. And tell them "don't shit me! We both know better and tell them what is actually going on" and ask them "what you intend to do about it?"
What I believe happened here during this conversation between President Johnson and Governor Wallace at least according to the LBJ tapes, was the Governor trying to convince the President that he was limited in what he can do to see that everyone in Alabama can vote. Because he governed his own administration, but not all of the local governments in Alabama. And what the President said was "don't shit me George! We both know who has the power in Alabama and that person is you. Now are you going to enforce the civil rights laws down there or, am I going to do that for you?