I could see LBJ sidelining McNamara much in the same way he did Hubert Humphrey especially as McNamara's views on the Vietnam war began to differ from LBJ's.
Where I could see it becoming very interesting would be in the 1968 primary season as McNamara, following LBJ dropping out of the race, advocating a "Vietnamization" of sorts (which he did in OTL around 67/68). With such a policy I could see him winning the nomination and the election to follow.
As President he'd probably serve one term before losing in 1972 to Reagan.