I am hoping that someone else will answer this question, but I can answer it in part. You have to remember that Johnson was from the South and that, until he won the Senate election of 1948, he was up for reelection every other year. Had he supported Civil Rights laws, he would have been voted out of power. Moreover, the entire Roosevelt Coalition that had guided Democratcs to control Congress and the Presidency during the years 1933 through 1968 (only one Republican candidate, the war hero Eisenhower, won the presidency during that period) depended on the rural Southern white vote. Indeed, after Johnson oversaw the passing of the Civil Rights bills of the 1960s, the rural white vote turned to the Republicans, the Roosevelt Coalition dissolved, and the era of liberal domination of American politics ended.

The Roosevelt Coalition does not tell the whole story I'm sure, but I'd argue that it is essential to understanding the politics of Roosevelt's self-identified heir LBJ.

Edit: I found this in Johnson's memoirs just now: ". . . I did not think there was much I could do as a lone Congressman from Texas. I represented a conservative constituency. One heroic stand and I'd be back home, defeated, unable to do any good for anyone, much less the blacks and the underprivileged . . . before I became Majority Leader, I did not have the power. That is a plain and simple fact."

On the same page, however, he also argues that "much of it [civil rights legislation] seemed designed more to humiliate the South than to help the black man," that this angered him, and that "these were emotions I took with me to Congress when I voted against six civil rights bills that came up on the House and Senate floor." -- all from page 155 of his memoirs Vantage Point.