ROME — When Stephen K. Bannon was still heading Breitbart News, he went to the Vatican to cover the canonization of John Paul II and make some friends. High on his list of people to meet was an archconservative American cardinal, Raymond Burke, who had openly clashed with Pope Francis.In one of the cardinal’s antechambers, amid religious statues and book-lined walls, Cardinal Burke and Mr. Bannon — who is now President Trump’s anti-establishment eminence — bonded over their shared worldview. They saw Islam as threatening to overrun a prostrate West weakened by the erosion of traditional Christian values, and viewed...