Ironically, I think you guys are seriously overestimating the negative effects of the Vietnam Non-War on an alternate Lyndon Johnson. Let's look at the four options available to Johnson once he became President:
1. Official intervention into North Vietnam in a conventional war. This was discarded IOTL as being too instigatory against the PRC, as it would put Americans right on the Chinese border. The Chinese had already set a precedent in Korea with what would happen if Americans were on the border, except this time the PRC was successfully testing nuclear weapons.
2. What happened IOTL with Operation Rolling Thunder and the Gulf of Tonkin Resolution: Expand American military involvement in South Vietnam, both on the ground and through bombing campaigns, but don't invade the North. Continue to operate under the (false) assumption that the Viet Cong were completely propped up by the North, rather than being largely domestic to the South, and therefore, as the assumption goes, eventually the North will give up and the Viet Cong will fold. This was supported by the vast majority of Johnson's advisors and military staff, and was, in other words, escalation from the status quo.
3. Keep it to the status quo, with American "advisors" acting as auxiliaries and supplemental forces for the South Vietnamese, but keep commitment to a minimum. This was George Ball's position, with his much more accurate memorandum of the situation in Vietnam than the information Johnson was operating from, and ran contrary to the 'common sense' of the time that escalation was needed to maintain American 'prestige' and the support of its allies.
4. Immediate withdrawal of advisors from South Vietnam. This option is so completely alien to Johnson's personality and American military/geopolitical philosophy at the time that we can discard it as akin to ASB.
Now, let's say the Ball Memorandum gets to Lyndon Johnson in time before he escalates the bombing campaign, and Johnson picks option #3, instead of option #2. The South Vietnamese inevitably lose, and Johnson can withdraw the "advisors" without much hassle. Johnson can write off the fall of South Vietnam by claiming that the only alternative was option #1, a full commitment and likely nuclear war with China, and that the former French Indochina was not a "vital interest" of United States geopolitical security. He can wheel out George F. Kennan, the intellectual godfather of the Containment policy, to back up his position. Since nobody at the time thought that a quagmire war was possible, the public at large would likely except Johnson's conclusion. Of course, he'll still take a hit in the polls, and the Republicans will go ham on the notion that the Viet Cong 'won without a fight,' but Johnson's savvy enough to minimize the damage without it resulting in the total and utter collapse of his Administration as some of you seem to imply. I think Johnson would likely win in 1968, whoever the challenger may be, but the main issue of the campaign would be domestic 'law and order' and desegregation, not how Johnson could have hypothetically saved a country 'most Americans couldn't have placed on a map.'