That it's a cheap political trick on the part of outgoing President Johnson! In fact, the pro-Nixon, pro-Republican sectors of the public will do much of the work for Nixon in this regard. Just like voters in Georgia recently made all kinds of excuses for the behavior of Roy Moore. He lost to Doug Jones in the Senate election, but just barely.You're assuming that releasing the recordings will actually cost Nixon the election. That's far from a certain result since the Nixon campaign is bound quite claim . . .
I do, however, think it would be risky for Anna Chennault or for Louis Kung (another Nixon emissary to S. Vietnam) to claim that it wasn't their voice. That's exactly the kind of specific detail the public can really bite down on, and find not very believable.
I like the timeline the OP sets up: Nixon still wins, claims of sabotage are leaked afterwards, pressure is increased on Nixon and on the South Vietnamese government to put together a peace treaty, 50-50 odds North Vietnam then overplays its hand.