In 1960 he was still publicly anti-Civil Rights, and an attempt to change his position during the campaign would just look like pandering.
How are you defining anti-Civil Rights?
His 1957 Civil Rights Bill defines him as being in the same place as Ike, the POTUS who signed it into law. Ergo he isn't worse off than Nixon in this area. Not by a longshot.
But unless he wins over those AA establishment leaders he did in OTL (which he was doing even as VP), and calls the King family when Martin Luther is in gaol, he's hardly likely to build up a superiority over Nixon on this issue.
But does he even need to show himself to be that much better than his opponent here? I don't see why he has to.
In a Johnson vs. Nixon race, Nixon is the television candidate.
I dispute this. Nixon's TV problem was of his own making, it literally comes down to him campaigning stupidly, screwing over his health, and not just to him being simply contrasted to John "God, why did you make me so beautiful?" Kennedy.
Debates aside, the TV appearances of Nixon '68 would have been competitive against JFK '60. His studio audience stuff, his convention speech, they showed him at peak form; IIRC
Reagan lauded the '68 acceptance speech as the best he'd seen.
There's a reason that Nixon wasn't able to build himself up in this earlier era.
OTOH, Johnson is renowned for having tamed his own presentational issues as soon as he got into the presidency in 1963--he didn't have that 5 years between the 'last press conference' and his declaration for 1968 that Nixon had to prepare in. (Though there was three years between Johnson's aborted convention 1960 appearances and his triumphant addresses to congress; still, that's a quicker learning curve than Tricky had.)
And while Johnson would have doubtlessly have campaigned tirelessly, he wouldn't have the same appeal that Kennedy had.
Kennedy almost lost the popular vote in 1960, and many people believe he would have lost the entire South without LBJ's barnstorming efforts down there.
Plus he lacked Kennedy's charisma and appeal to Catholic voters.
In this scenario the Catholic vote in 1960 has a choice between Nixon and LBJ, not Ike and LBJ.
Anyway, considering JFK ran four points behind the Dem nationwide congressional tally, it's a good bet he didn't win enough extra Catholic votes to make up for the Protestants he lost in the prez ballot.
Being a cross-denominational candidate is what is needed to win 50%. JFK unfortunately didn't hit 50% PV.
Johnson would have carried the entire South and would have done very well in the West. Look to Truman in 1948... Rough Johnson victory map
I doubt he carries the Mountain West, but I more seriously also doubt Nixon can build that solid midwest/northeast firewall you give him. This is the first POTUS race coming out the 1958 recession, after all.
LBJ had crazy alliance building skilz with labor, and this despite him voting consistently for Taft-Hartley type laws in the years after WWII. I also like to go with the fact he wrote a liberal GE manifesto with the help of the Reuthers when he was considering running in '56.
Also, he was seriously good with ethnics. He was a Philo-Semite, though Nixon should probably be lucky enough to avoid having this contrast with his own attitudes ever becoming apparent...