I tend to think peace in Vietnam early in 1969 might have been possible, just like peace in the 'Great War' might have been possible in 1916.
Difficult, but still possible.
Don't try too hard but keep trying. And hire additional negotiators as necessary. In poker terms, pick up the cheap pots no one else wants (the unexpected opportunities).
Aside from the logistical and bureaucratic problems, the political reactions in the US from the right wing if Hubert Humphrey were to do such a thing, the possible ramifications with initiatives with the Soviet Union and China, and the fact that ARVN needed to be trained to take over American positions, there is also the fact that neither side in Vietnam had any incentive to negotiate in 1969. Why would they? It'd be like asking Thieu and his nation to commit suicide to help the doves personal anguish, basically, and Hanoi basically figured they could go for victory and bleed the Americans-or invade. Neither set of Vietnamese WANTED a realistic peace agreement in 1969. Also, there is the whole geopolitical situation in the region.
The Prague Spring had just happened without an American response with the US looking weak as a result, and there were revolutionary elements in the West who-it seemed like-could tear the place apart. Worldwide, the Chinese and the Soviets were about to undergo near nuclear war, and there was also very turbulent "rest of the world". The Soviets had reached nuclear parity with the US. A weak-or perceived as such-America wouldn't be able to do anything about it. And the domestic scene... George Wallace and "Bombs Away" LeMay had won a nontrivial set of the vote. As Lyndon Johnson said-"Ignore those little shits on the campuses. The great beast is the reactionary elements in this country, they are the ones you have to fear." This was in 1965, at the height of his power and prestige.
This would be the case with HHH even moreso than Nixon, he could not afford to just capitulate even if he could, both domestically and internationally. I don't think anyone in office could have contemplated unilaterally withdrawing like that.
I know you should never say never, but I just don't see it happening. Obama has faced some of these problems with Iraq and Afghanistan. These sorts of things take time. Humphrey (or RFK) probably would have perused Vietnamization or some plan like it to at least stop the American bleeding-whoever is in office won't have a crystal ball, and that is probably what they will do. The idea we are getting out in 1969 is as fantastic as the fantasies of the right wing, who wanted a crusade. Only those without power can cherish either extreme.
I also, as a side note, think its a little much to expect the new administration to do in six months what the previous two Democrat administrations had not even thought of or could do over eight years, as many of former Kennedy/Johnson people seemed to expect OTL when Nixon took office, and perhaps also would if Humphrey does. Suddenly became doves and suggested concessions that they never would have implemented themselves. Many of them acted like innocent bystanders, or started blaming the current administration. This might be lessened, but it wouldn't go away, with Humphrey, I think. Similar things have happened with Obama attempting to clean up the GOP's mess in Iraq, and the GOP line to foreign policy shifting to opposing Obama even if he did what normally would be thought of as conservative.