I've thought about ATL 1968 a fair amount myself.
Given the OP premise, clearly this is a fairly late POD in the year of 1968, since LBJ pulling off such a coup as a peace treaty early in the year would probably mean he himself runs for reelection and I would guess, has a good chance of winning in his own name, albeit not the landslide victory he did win OTL in 1964.
Things changed fast that year and it is indeed realistic the opportunity for a peace treaty would only come too late for him to pursue the Democratic nomination. I am interested in ATLs where someone other than Humphrey wins the nomination, but again given LBJ's power in the Democratic establishment, such ATLs are long shots--only Humphrey would be acceptable to Johnson, and only after his own chances were torpedoed. So the POD has to be fairly late.
Now then, is it realistic for Hubert Humphrey to be the one to call out Nixon's quite illegal antics? I would say no, not at all. "The buck stops here" said Harry Truman, as elected President, it was Johnson's job to do this. The reason he did not OTL was that he was able to know what Nixon was up to by means that were illegal themselves. Johnson was playing the game of what passed for ethics in the Cold War era. To state the credo as I understand it, it was, among its believers, that the government of the United States was fundamentally and axiomatically legitimate because it stood for the true interests of the democratic public that elected it, and as it was engaged in an existential struggle against a ruthless enemy with no regard for human rights or fairness, it was often necessary and thus proper for these officials entrusted with advancing and protecting American interests to do things that conflicted with the more high-minded statement of the principles we were supposed to be for. The President had to have the free hand to do stuff that was in fact illegal, if they did it in defense of fundamental American and human freedoms. But this meant keeping official silence about such maneuvers, and that gave Nixon the cover he needed; calling Nixon's foul would also disclose the foul of the administration spying on private citizens by means that had no legal defense. If Johnson had judged that it was necessary and proper to blow the whistle on Nixon's maneuver, he surely would have taken doing that on himself, especially in a situation such as late 1968 where he was already going down in flames personally--better to keep his proxy candidate's hands clean, as well as the Constitutionally straightforward thing to do.
I think Nixon most certainly should have been blocked somehow, and perhaps there is another way it could have happened. Nixon's scheme depended on a critical number of South Vietnamese officials uniting to telegraph they would not cooperate in supporting a treaty hashed out between Johnson on one side and Hanoi and Moscow on the other, and figuring they would get a better deal if Nixon were elected to better defend their interests as future US President.
Rather than exposing Nixon (which would certainly be the moral high road to take, and perhaps as I believe, better for the nation to shine a spotlight on the degree to which the practice of "defending freedom" involved the practical subversion of it) could Johnson have instead have played harder ball with the South Vietnamese politicians, and made it clear to them that on one hand, he would make sure the South would be protected and their personal positions secured no matter what the treaty said--but on the other hand if they did not cooperate with this political necessity, he would see them deposed as necessary? On paper, the United States was providing aid to a freedom-loving republican state that was being subverted by ruthless enemies of freedom, but in fact the whole Saigon regime was pretty much a construct of US power, and these privileged gentlemen owed their positions, privileges and safety to Yankee patronage. To purge them would be costly on many levels, but it would hardly involve a greater violation of such high principles as sovereignty or legitimacy than putting them there in the first place had!
If it had been easy for LBJ to take such a step, I suppose he would have OTL. It is thus clearly a long shot, but I think less of one than the bigger political risk of blowing the whistle on Nixon publicly and disclosing how he was aware of Nixon's (quite criminal to be sure) antics.
Now, if we can suppose that one way or another Johnson can counter Nixon's move and thus get the treaty signed before the election, or even after it, I do think that while clearly this would not truly end the conflict in Vietnam, it would change the terms of it drastically. OTL the treaty achieved by Nixon was a step on the road to the total defeat of US intervention in southeast Asia and the victory of Hanoi, but I think it would be different if accomplished earlier.
IMHO and understanding, the treaty Lyndon Johnson was apparently near finalizing with the North Vietnamese government and their Soviet patrons, was essentially identical with the treaty Nixon agreed to in 1972, so if the latter is deemed acceptable in any sense, the former would be equally so, whereas anyone denouncing LBJ's version as a joke or the like (essentially surrender of the Southern Saigon regime, "Republic of Vietnam" to Communist rule) would be admitting Nixon made the same surrender, ultimately. I actually think that the overall situation depended very strongly on shifting attitudes in the USA as a whole, and that the same treaty agreed to effective the start of 1969 might give the Saigon government better odds of long term survival.
Basically, already by late 1968 the Vietnam war was widely unpopular in the USA as a whole, but stretching the war out another 4 years, despite the fact that Nixon did lower the rate of specifically US force casualties by various stratagems, greatly deepened and broadened the unpopularity and led to a Congress more deeply opposed, with both newly elected Members and long-term incumbents looking at constituencies more strongly opposed than ever. Whereas in 1969, in the wake of a treaty and sudden and major drop in US losses, old school Cold War mentality might have better prevailed, if not in the general populace, at least among their incumbent Congressional representatives, while Johnson managing to accomplish this diplomatic coup would go a long way toward redeeming the general reputation of the Democratic led bipartisan establishment. Of course among hard-core anti-war activists this would not make a big difference, but the difference between 1969 and 1973 in terms of the more widespread and deep US mentality and their Congressional representation is a matter of deeper fatigue of the segments of the population that supported the war in the first place. A certain core of various segments never wavered and even doubled down in the face of popular rising skepticism, but what mattered I think was the erosion of support in the middle ranges.
Thus, in the event that the outcome of a treaty essentially identical to the OTL one being identically violated by the Northern PDRV forces making a conventional military invasion of the South, it would be possible for any US President of either party to call for emergency reinforcement of the Southern RVN forces. Any treaty provisions forbidding Uncle Sam to aid the South overtly would be out the window with Hanoi violating the treaty so egregiously, so it came down OTL to the US establishment being unable to muster any credible commitment to defending the South in this manner by 1975. If the same thing were to happen in 1971 with Hanoi and the Kremlin having signed off on an end to the conflict in '68, vital support that OTL was eroded by what IMHO can be described as 4 years of bad faith by the OTL Nixon Administration would be on hand--a segment of the population would still be dead against re-escalation, while meanwhile the danger of the conflict triggering global thermonuclear war would be a chilling factor to be sure--but against that, the Soviets would not look very credible diplomatically arguing that the oppressed people of Vietnam were being bullied by the Western capitalists, if in fact it were the Northern forces who attacked the South openly. The Soviet regime was at the end of the day rather risk-averse and first of all I think they'd have strongly discouraged gung-ho factions in Hanoi from proposing such a move, encouraging their clients to think of the treaty as a win allowing them to go about building up the North on their own terms without devastating costs from Western strikes against them, and clandestinely maneuver for influence in Laos and Cambodia as well as in the South as best they could. And while we can't dismiss the possibility the Northern rulers would disregard Soviet urgings as they had done before and make an impetuous move anyway (considering Southern weaknesses, more below!) I do think it might be in cards for Moscow to disown them, or anyway limit their demands to restoration of status quo ante, with the North remaining Communist, but voice no opposition to Western allied forces (mainly US, but with considerable aid from Australia as well as what remains of RVN domestic force, and nominal support of SEATO--Australia, but also Thailand, the Philippines, Taiwan (assuming the US has not as OTL shifted to recognized the PRC, which we still hadn't OTL as late as 1971), Japan, Malaysia, Indonesia, and Singapore) concentrating to clean the clock of any intruding Northern forces and even pursuing them past the border into the North, as long as when the dust settles the North gets back control of its pre-invasion attempt territory. And perhaps Moscow would sign off on the North losing some territory in the middle too.
Even with nominally no US forces in South Vietnam whatsoever, I think there would be a fair number in the region, largely in the Philippines and perhaps in Thailand, and given a major diplomatic shift, perhaps Cambodia and maybe Laos, as well as in Taiwan and Japan. US air support of an initial ARVN defense of the South could be authorized in a matter of hours and deployed to RVN bases in not much more time, considering that Yankees had constructed these bases and that the Southern Vietnamese air force would be operating pretty recent US models donated to them. Therefore I think that, given US willingness to defend the South against such an overt invasion as prevailed OTL, the North would not calculate a good chance of victory by such means and would not commit to such a foolish venture.
Now on the other hand, OTL the 4 years of delay before Nixon agreed to essentially the same treaty Johnson was seeking did make a difference in terms of the Saigon RVN regime consolidating political control of the South, largely due to massive attrition of activist pro-Communist forces among the Southern citizenry itself. Now this was due in large part to means I cannot agree were what I would call legitimate, involving as it did police state terror and ruthless violence--which is not to say I think Northern methods were less so, but it does render the notion that we Yankees were there to protect the freedom and human rights of the South Vietnamese rather hypocritical to say the least. And this police state portion of achieving Southern "security" and "legitimacy" was very much something Uncle Sam was involved in, being essentially a CIA project. In 1969, much of this political "cleansing" was yet to be accomplished, and so with a treaty identical to OTL in place, the North would be pursuing aggression counter to the spirit if not the letter of said treaty by means of continuing to seek to support subversion in the south, rather than considering overt military invasion--though to be sure, if anything I suppose the forces Saigon controlled on paper were less reliable and less competent than they proved to be in OTL 1975! The crucial difference is that the South would remain under Yankee protection on similar terms to the North being under Soviet (and to an ambiguous degree, PRC) protection--playing a war game on paper with Vietnamese forces only would be a fantasy in the ATL 1969-72 period, as it was not in OTL '75.
But certainly overall Saigon was weaker by itself in 1969. There were other factors besides the general political cleansing practiced so ruthlessly OTL during the Nixon years. South Vietnamese society was transformed over that time to be much less centered on traditional rural villages, much more concentrated in urban areas which in turn were subsidized by Yankee "aid" in various forms, which to be sure primarily enriched just a stratum of the national population directly, but anyway did change the overall social calculus to undermine the channels whereby the combined domestic grassroots and Northern cadre resistance opposed the factions the US favored in Saigon. If the North could invade with a free hand as OTL, their victory would be more assured earlier than later I suppose.
This is academic though if in fact US leadership would assuredly sweep in to defend the South; perhaps considerable territory might be lost before American and allied forces could stop an invasion, but having stopped it, US forces in 1970 or so would be competent to wage and win a conventional set-piece ground war of the WWII type, without even having to resort to threats of nuclear strikes-indeed air power alone, using conventional bombs but much augmented by Cold War era technology, would be far more effective (whereas the Soviets, willing as they were to supply Hanoi with defensive arms and even some clandestine Soviet personnel perhaps, would I predict wash their hands of invasion--domestic PDRV airmen were competent and devoted, but in the air I am confident that the combination of US sophistication and material resources would rapidly secure Southern airspace and put the North on defensive fast). To inject a personal note, my own father flew F-105 fighter-bombers, in a bomber role, out of Thailand in 1968, and remarked that this "light" supersonic jet bomber carried more tonnage of ordinance than any "heavy" bomber of WWII could. More infamously, overall the tonnage of bombs dropped by US forces on Vietnam and neighboring countries once the war spread to Cambodia and Laos did indeed exceed that dropped by all sides in World War II, OTL.
Given such air cover, I suspect the ARVN would rally and do much to blunt the Northern strike, and soon after major US and allied forces would arrive to reinforce them and drive back the invaders--whether they would stop at the nominal border is a question to game out in detail, but I would think by then the post-war treaty would already be hammered out and signed by both Washington and Moscow, and Hanoi would know they had best agree to a truce or be cut to ribbons when left out to dry by a Soviet regime with no interest in World War Three.
So IMHO, it comes down to US resolution, and avoiding 4 years of bad faith carnage (however mitigated in terms of US force loss rates) and political hypocrisy would make a big difference in outcomes. Indeed I think the Northern regime would wisely avoid the path of overt invasion, whereas no treaty would prevent the US from shoring up Saigon in ways that do not translate directly into countable deployed military force but do manage to secure the southern regime (in ways I personally think would be deplorable and worse, with effects I don't think would be very nice on the southern population overall, to be sure) politically against domestic insurrection.
It is conventional wisdom that the domestic southern opposition was largely decimated by their failure to take over in the Tet Insurrection--OTL this proved a Pyrrhic victory for the US/Saigon side because in context of the war going on another half decade, it undermined the "light at the end of the tunnel" mentality of the powers that were in Washington. In an ATL where a treaty of peace is signed within 12 months it would be different I think. Popular culture portrayals of the shifting political sands in the USA point to the discomfiture of these authorities at that moment, but in context of a diplomatic victory soon after I think it would be different in the USA. In such an ATL, the fact of decimation of genuine Southern domestic opposition would offset the weakness of the Saigon government in the context of ongoing resolution by US authorities to prop them up. As noted I don't think the necessary means to prop them up would be what I would call legitimate but they'd pass political muster despite an energized countercultural opposition in America denouncing them. End the involvement of drafted US conscripts being sent into South Vietnamese villages to conduct terror campaigns, preempt such iconic events as My Lai (never mind such things had already happened--the politically if not morally important thing is to prevent more of them in the context of ongoing US grassroots skepticism worsening), end escalating bombing campaigns, leave the ongoing war to be what the North could clandestinely orchestrate versus what the ruthless if not very well politically grounded Southern authorities, with clandestine US backing via CIA volunteers and purportedly civil and peaceful aid to the South economically continuing to counter it, and the RVN could stand, and the North would not dare their OTL victorious endgame of overt invasion.
Seeing that others have responded to the OP in the time I have been writing this, I will pause speculation on how this impacts US and world society and history generally outside of the narrow matter of the situation in Vietnam itself to see what others have to say about it. These are the terms in which I see a form of the OP stipulation being realistic and the immediate bearing getting the treaty passed in 1968 (or very early '69, before Nixon takes office) would have on the Southeast Asian front. Even if LBJ could not pull it off before Humphrey loses the election as OTL, getting it passed before Nixon takes the helm would change many things, all for the better from Johnson's point of view.