’s bye-bye, Beltway for Cory Booker. Newark’s mayor, who was gunning for a spot in President Obama’s Cabinet, lost the chance after he shot his mouth off during a blunderingly honest TV appearance last month, sources told The Post. “He’s dead to us,” one ranking administration official said of the prevailing feelings at the White House and Obama headquarters in Chicago. Booker had been angling for the housing secretary gig in a second Obama term, according to sources in the administration and close to the mayor.