The problem is that neither of them had a reason to reconcile. From Dallas to late 1964 is when RFK was depressed over JFK's death (the outpouring of support from his inner circle and New York voters returned him to his old self personality-wise by January '65), and LBJ kept trying to extend peace feelers. Most of the feelers were misinterpreted or outright ignored. Once when LBJ offered an invitation for a drink and a swim, RFK told an aide: "... he always invites me to come over for a drink or a swim and discuss our issues. I have no interest in becoming involved with him." Even with intimates, Bobby didn't like and wasn't comfortable in what we'd call an "Oprah situation", namely an emotional confessional. On LBJ's side- he was paranoid and thought that RFK was constantly undermining him (Ted Kennedy recalled RFK saying: "for Chrissakes, I'm the junior senator from New York, he's the POTUS

"), so he did the same. Torpedoing his bills in the Senate via a few phone calls, purging bureaucrats and the DNC (DNC purges proved fateful to HHH in '68 IOTL) of real or imagined RFK loyalists, many of whom were better-quality personnel than his own loyalists. One of the few who survived that was Pat Moynihan. Also, LBJ knew that RFK wanted the WH- his ambition for it equalled that of both Clintons.
Here's a picture that sums it up.