I'm not at all sure how the nation reacts to an arrest of a strong rival of the established party, the leadership blamed for many messes besides Vietnam (many of which I would not think were messes, but some people would) so very close to the election. The suggestion to change the year, essentially Johnson helping himself to cutting Humphrey's term in half, might mitigate things a bit, but it is more radical change to irritate the "Silent Majority."
Remember LBJ himself cannot be reelected. He is not on the ballot, Humphrey is. OTOH he gets to manage the transition to whoever is elected.
If Johnson could ram through a conviction before election day, then Nixon's status as a valid candidate is in doubt. But if an arrest and publication of charges looks self-serving and dictatorial, imagine what such an apparently kangaroo court ruling would look like! As long as Nixon is not actually convicted, which with usual US trial procedures would take months, he is a legal and valid candidate for the office of the Presidency. People might therefore vote for him to defy Johnson. They might convince themselves (wrongly, but easily) that the treaty Johnson is negotiating is bad, and should be torpedoed, and calling Nixon and Anna Chenault heroes, and endorsing the stand of Thieu and the other South Vietnamese leaders.
It is my personal belief that if Nixon had not muddied the waters, although the Saigon government did strongly dislike the process and feared the outcome of a peace treaty, they also understood that they were dependent on US good will to survive at all, and knowing there was a good chance they'd have to deal with Humphrey after next January (and of a certainty they had to deal with LBJ until then, barring his resignation, impeachment or assassination--then they'd be dealing with Humphrey early) they would sullenly and grudgingly follow the US lead. Thus the significance of Nixon's action was not to put the idea into Southern heads that they didn't like the treaty--it was rather to send them a message that if they only took a bold stand for what they wanted to do anyway, they would be rewarded. The Democrats would be embarrassed and gotten rid of, from Presidential power, and the new President being a famous anti-Communist would surely have their backs. This was the message--and it was clearly treasonous, in the sense that he was working to frustrate current Administrative policy in foreign affairs.
If I am right then Johnson would have been able, if not before the election then before leaving office, to hand the nation "Peace in Our Time" in Vietnam--and ironically, as I have argued elsewhere, the USA under whatever Presidential leadership would be both more able and more willing to guarantee the South against an overt military invasion of the kind that actually caused the Saigon regime to fall OTL. In short Thieu, or whoever might succeed him in what passed for due process in the South, would keep his job after all, because the North would not invade knowing the Americans could and would intervene to stop it. Elsewhere I've enumerated what LBJ's motives and wishes were at this point; they went beyond swinging the election his party's way.
So--what happens if Nixon is thrown into jail, accused of violating the laws that forbid private citizens to work against the intent of US foreign policy for private gain (or any reason at all, no matter how lofty), Nixon then wins the election (perhaps more closely than OTL), then when the prosecution lays out the evidence, it is damning? Might the Republicans ask Nixon and Agnew to both resign before taking office, or make it clear to them that if they are allowed to be sworn in it will only be on the condition they then resign immediately?
Then again, why pick on poor Spiro Agnew? The idea that Agnew was a candidate was played up as a joke, as an instance of Republican irresponsibility much as the choices of Sarah Palin or Dan Quayle were treated in their respective years--but Constitutionally speaking, he is Nixon's legitimate successor, and if he could not be shown to have anything to do with the Vietnam scandal, the deal might well be--Nixon bows out, before or after the Inauguration, in favor of Agnew, who would then pick someone else to be his VP.
Thus Nixon, as a private citizen with no Presidential immunities, faces a trial for his interference. A trial he could conceivably win. In fact, the Republican deal with Johnson could be made entirely contingent on Nixon's trial outcome--say they agree to let Johnson continue to serve as President, under close Republican supervision, until Nixon's trial is settled, and if Nixon is not found guilty he then takes office; if he is, Agnew then takes office.
Or the matter could be settled as a plea bargain--in return for charges being dropped, Nixon bows out, no harm, no foul, maybe he has to also agree never to run for public office again.
But meanwhile, what about the Treaty? I think if Johnson arrests Nixon and the charges are made public, then the South Vietnamese leadership will not cooperate with the treaty process. They want to see Nixon vindicated; perhaps their words and actions might make his legal case worse, but they are interested in the political case anyway. They will screech and scream that Johnson, the nogoodnik pinko liberal, wants to sell Vietnam into Communist slavery to get rid of a problem, and Nixon is their brave hero standing for freedom-loving South Vietnamese. The treaty is dead and Johnson's choices are to give these ungrateful Saigon mandarins whatever they want on their terms, to capitulate to them, or to cut off aid and let Hanoi take over for real, thus appearing to make their case for them. Or of course he could do as he, and Kennedy, and Eisenhower had done before and topple the current leadership and look for new puppets who could better be hoped to toe the US line, and go on sending more money and more men into the Vietnam rathole, which would be Agnew's mess to clean up as he sees fit after January.
Johnson's dilemma is that getting the treaty is his goal for reasons far more important than who wins the election, but by taking the action he did OTL Nixon guarantees LBJ will fail, and cannot vindicate himself.
Even from a jail cell, Nixon wins. He wins even if the election goes the other way, and Humphrey gets the electoral votes, or the election goes to the House.
And no matter how solid the evidence and how exemplary a trial Nixon gets, if he is convicted there will always be those who disbelieved the charges.