In the '50s, Nixon and the Kennedys got along. Nixon then felt personally betrayed by Kennedy for several reasons, and no one was more implacable in the face of wrongs, real or imagined, than Nixon. I personally don't think his mental makeup would have let him go blithely along with a hoax if there were a chance to convince people they should just like him more than they liked JFK. We know from Watergate that Nixon didn't necessarily worry as to whether things were in his best political interests or not once he'd decided people were against him. (Continuing to plan to use the IRS or the FBI against his enemies even after being under investigation for same, for example.) We also know that he was willing to do things that might make people see him as "soft on Communism" (see "Only Nixon Could Go to China").
The thing is, though, it's still hard to know what Nixon would or wouldn't do, because Nixon was in many ways a very private person. I read a book about mental illness and figures of political power, and the author suggests that Nixon was perfectly mentally healthy. I, however, suspect he may have been a low-level paranoid schizophrenic. Certainly the paranoia that author chalked up as "well, you know, people really were out to get him" doesn't hold up when you really start looking at "the enemies list." (Seriously--one of the twenty people on the original list was Paul Newman.) I think he would have exposed Apollo, and for two reasons. One, "make them like me by showing them that Kennedy lied to them." Two, I really do think he would have thought the hoax was unAmerican and unpatriotic, and I think he would have known that it could not have stood forever, given our enemies--like the USSR and Paul Newman--a chance to crow over it. I don't think he would have seen it as triumphing over the Commies, because it wasn't real.