This is not implausible: he could stump for Kennedy and use his position in the Senate to exert leverage to get his own agenda accomplished in exchange for helping Kennedy. Indeed, I could see Johnson trying to eclipse Kennedy from the floor of the Senate. That would lead to on-the-surface cooperation but the smiles and back-slapping would be phony as all hell. It could even lead to intramural warfare for the '64 nomination.
Sooner or later, the Bobby Baker affair will surface. The Kennedy-Johnson relationship is critical at this point. If it's approaching polite/cordial, Kennedy may flex some muscle to downplay it and force Johnson to owe him. If not, Kennedy takes a high road (i.e., throws Johnson under the train) and indulges in the Schadenfreude of seeing his frenemy go down the tubes as a two bit grifter. Of course, that could backfire: southern Dems would try to oust Kennedy in revenge.
Here's where it gets particularly byzantine, if not outright ASB: constantly fending off southern Dems, Kennedy has to turn increasingly to the GOP for help. One evening, in a private conversation over bourbon, Barry Goldwater says jokingly that Kennedy has spent so much time with the GOP, he might as well change parties. Kennedy responds that what he says isn't half bad. Goldwater grins and says he was kidding. Kennedy goes full Jack Webb and says he's serious.
In a matter of weeks, fed up with the intransigence of southern Dems, Kennedy pulls an historic first and changes parties. His old man, who supported McCarthy and vowed to back Nixon if Jack didn't get nominated, buys in but Bobby does not (here begins Kennedy intramural warfare). Richard Nixon is chagrined to say the least but is significantly mollified when Kennedy gets Dean Rusk to resign, and is himself made SecState.
Whew...I'm exhilarated but tired just thinking about this! 8-)