We could avoid tensions with a successful conspiracy of the pure coup while Eisenhower is in office and Castro most likely would get elected when democracy gets restored, preceding Chavez (that is, if the guy doesn't butterfly away) nearly half a century before. I want to try this in my timeline. I can imagine Guevara would be livid.
The problem was Castro was going to 'go-Communist' no matter how he got into power so it would likely be even MORE of an issue if he were first 'elected' and then turned Communist. Even if we assume he doesn't go full blown Communist, (and with Guevara around that's not likely) he's going to be far more socialist than the US is likely to be comfortable with. If you can make it work I'd love to read such a time-line but a LOT of the inner-circle of the rebels simply did not trust 'democracy' anymore and were not actually willing to trust putting it back into place.
Which brings me around to something else I'd wanted to touch on; if an invasion or conflict in Cuba can be avoided then Nixon has a chance to address a concern of Khrushchev's that was eventually "resolved" OTL through going to the brink of nuclear war and then stepping back, whereas I think Nixon and Khrushchev could avoid that alarming step.
Eisenhower had reluctantly put IRBM's (Jupiter specifically) into Italy and Turkey which essentially ringed the USSR with very low warning time systems that were arguably best suited for a surprise First Strike scenario. He had not favored even developing IRBM's seeing them as disruptive and taking effort and resources away from ICBM development. But Congress lobbied for them and essentially demanded they be 'deployed' once built and since England was getting the duel-key Thor set up, Italy and Turkey were the next 'logical' deployment areas for such weapons. Of course Italy and Turkey knew this and were less than thrilled with the idea of 'hosting' weapons that would make them targets for nuclear attack. Once enough ICBM's were coming on-line the Jupiter's made a lot less sense but the incoming Kennedy administration had run on a "Missile-Gap" strategy and therefore were not really keen on being seen as even considering 'removing' active missiles from service so they didn't address the issue till after the crisis. Nixon on the other hand still has enough cred to make it plausible that when he meets with Khrushchev and the issue comes up, (and it will) both of them can walk away from a conference touting an agreement to remove those missiles for a similar form of 'trade' with the USSR and both of them come out in a much better position than the aftermath of the OTL crisis.
It's possible that even with an US invasion of Cuba there could be an opening to use the removal of the missiles as an 'olive-branch' to the USSR which will likely be accepted and Nixon could probably 'spin' to the US and West.
Randy