Michael Moore sparked a Manhattan film festival rivalry when he showed up at a “50 Years of the New York Film Festival” screening series wearing a Tribeca fest ballcap. The documentary maker arrived Tuesday night at a Film Society of Lincoln Center screening of his 1989 “Roger & Me” in a hat from Robert De Niro’s downtown festival. “He showed up wearing a Tribeca hat because he’s on a jury for them this week,” said a spy. “He profusely apologized, and theater staff quickly found him a hat with the Film Society of Lincoln Center logo.” Moore then popped on the Lincoln Center hat, saying, “This fits perfectly.” He told the audience that George Clooney used his doc as a dating tool. A spy recounts, “[Moore] said Clooney told him awhile back that, when he had dates, he would watch ‘Roger & Me’ with them to gauge whether or not they ‘get it.’ ” Clooney was married to actress Talia Balsam from 1989 to 1993, who presumably got it. Perhaps more recent ex Elisabetta Canalis did not.