Johnson would have privately voted for and supported Nixon.
I do not believe that. As much as LBJ hated RFK, I believe that Johnson would have given RFK an endorsement. LBJ's great liberal legacy was Civil Rights, and he wasn't going to trash that for history.
LBJ endorsed McGovern in 1972. I believe that doing so helped the way he was viewed by liberal historians, and why his ranking amongst Presidents has risen over time. If he had endorsed Nixon, he'd have been seen as rejecting the positive part of his legacy, and he wouldn't have had the upward reassessment as a President that he has posthumously experienced.