(I didn't intend to leave out a reply to David T, but I burned myself out typing)
No Republican ticket could be strong enough to make Johnson do that, especially in 1964 when he has all the cards, post assassination and during the Civil Rights Act. Johnson would not be so intimidated of Romney that he would do that, not when he is desperate to show that he is his own man. This is just flat out ASB.
In 1968, Johnson probably would have (and could have, at least in Texas and probably elsewhere in the regular Party) tried to sabotage Bobby if he ran against Nixon. That says it all.
Hindsight is 20/20. We have hindsight, and we know how things did turn out and we know the factors in play, and we can say the Democrats are most likely to win 1964 regardless of any other factors. However, that hindsight is not something Johnson has.
It seems every president, including those who won in landslides, were worried about their chances for reelection and not sure if they would be reelected. That's been the case with presidents from Lincoln to (famously) Nixon, even though history shows they were comfortably reelected.
We have the narrative that Johnson has inherited the White House from Kennedy, and Johnson will continue what Kennedy began before being sadly murdered and honor his legacy. But we could have had the narrative that Johnson is not John F. Kennedy, that he looks to Americans like a Southern conservative ticket balancer who may be open to carrying through what Kennedy wanted, and he has not legitimately attained the presidency. It could be a narrative that 1964 would have been a clincher for John Kennedy, but now its an open question to who will succeed him. And regardless of the realities of that narrative, that could have been something playing in Lyndon Johnson's head.
Bear in mind as well, we're opening up the possibility of different things happening after Kennedy's assassination. What Johnson does is make sure he legitimizes himself as Kennedy's successor to the American public, and shows that he's a Liberal who will carry out the domestic agenda, including Kennedy's Civil Rights bill. He does that, meaning he doesn't have to run with Robert Kennedy - something he also does not want to do also because he does not want it to look like he cannot win the presidency on his own, nor does he want to live in the shadow of Kennedy (despite him campaigning heavily on Kennedy's image). Imagine, though, if things falter, such as if Civil Rights legislation doesn't pass before the election.
The Republican nomination of Barry Goldwater is another component. Goldwater does not have a snowball's chance in hell, he's alienated the Republican moderates and the rest of the Rockefeller wing, he's not going to win many others over, and he only won the nomination because of a Conservative zealotry and insurgency in the Republican party. And he's the opposite and the opponent of the Kennedy/Johnson policies. There's no concern about Goldwater winning, and Johnson can really take command of that election, and he can do as he pleases. A moderate Republican is a whole other entity. George Romney looked like a legitimate threat to Kennedy, and I imagine he would look the same to Lyndon Johnson. I imagine Johnson may fear he'd capture voters if Johnson did not prove his legitimacy to the voting public, that he'd get voters who liked Kennedy but did not necessarily like Johnson, that he'd get voters in the North as LBJ was a Texan, etc.
And that's where the hindsight issue comes in. We know Johnson would have won 1964 regardless of what he did. Johnson doesn't know that. He may know that against a Barry Goldwater, but he wouldn't against Rockefeller, Romney, Scranton, etc. So that, plus the possibility for him to not do as well as he did in the OTL leading up to the election, is why he may feel compelled to run with Robert Kennedy. And we do know he thought about it, even though he thought of it as something he didn't want to do but may have had to do.
If Romney is the nominee, the South will not go Republican. Romney was strongly in support of civil rights, he marched with civil rights protestors. George Wallace will run a third party campaign and sweep the OTL Goldwater states at least, maybe even do better since he isn't a Republican. Strom Thurmond might not even defect in this TL.
The issue is not just the issue of race. The issue is also that the South is getting more white collar and beginning to lean Republican. Look at the election map for Eisenhower in 1952 and 1956 and Nixon in 1960. The Solid South is becoming less solid, and the region that is Solid is the Deep South. And that region is even more anti-Black, anti-Civil Rights, and is likely to go to Wallace.
Beating Goldwater would be hard but not ASB. Wallace runs as a third party. He loses to LBJ nut not in a landslide.
Barry Goldwater almost did not run in the OTL. In the wake of the Kennedy assassination, he did not want to run for the nomination anymore, but he was convinced to do it.
That's really an interesting thing for me that's coming out of this thread: the 1964 race of the OTL was pretty close to what it would have been under Kennedy. This scenario opens up a lot of possibilities for things to be much different than they would have been under Kennedy.
Not to infer that it has to be how Romney gets the nomination, but Kennedy's death could have gotten Goldwater out of the race, opening the door for the moderate Republican. As a result of a moderate Republican, there's a strong chance of Wallace running as a third party. That affects the history of the eventual Democratic split and the political evolution of the South, and we could see things unfold much differently. As a result of a moderate Republican, you have the possibility of Johnson feeling he had to run with Robert Kennedy, with all that entails.
EDIT:
My ideal scenario would be for Romney to win the nomination as a compromise between Rockefeller and Goldwater. Maybe have Goldwater drop out for a while and Rockefeller lead the field, before Goldwater coming back ala Ross Perot and acting as a spoiler, with Romney coming through as the compromise.
The pity of it is, if Goldwater had not won the 1964 nomination, it would have probably gone to Rockefeller regardless. And he would have lost (his divorce and remarriage would have been a major issue), and probably used it as a springboard for the 1968 nomination and election, again shutting out George Romney.
