Well, we might not be able to totally kill off Sino-American rapprochement with a gaffe or two during Nixon's trip, but maybe we could still put in a few dents?
For example, what if when Nixon gets to China, he never meets Mao? Mao's health at the time was pretty dicey, and perhaps he takes a turn for the worse in February 1972, and his doctors decide that meeting Nixon could kill him?
So then, Nixon arrives to a very tepid and low-key reception in Beijing (in OTL, the White House and press corps was very surprised and disappointed that Nixon wasn't given the same red carpet treatment that many other foreign leaders were given in China -- no cheering crowds or tickertape parades and such), and then they ferry him around to some tourist sights, have a few banquets, issue the Shanghai Communique, and Nixon goes home, but without that iconic photo of him shaking hands with Mao.
This certainly couldn't kill off rapprochement, and it will still be seen as a diplomatic accomplishment of Nixon's, but still, everyone (and probably Nixon in particular) will likely feel like he had been snubbed by the Chinese, and perhaps a few people might buy into the right-wing's complaint that we rolled over for them and got nothing in return.