WASHINGTON — As Donald Trump rolled out the red carpet for a parade of foreign dignitaries in recent weeks, some aides to President Joe Biden took notice — and umbrage — at what they saw as the former president playing pretender-in-chief. In less than two months, Trump has hosted Polish President Andrzej Duda, Hungarian Prime Minister Viktor Orban, former Japanese Prime Minister Taro Aso and David Cameron, the former British prime minister who now serves as the U.K.’s foreign secretary. He’s also talked with Saudi Crown Prince Mohammed bin Salman and others by phone. It’s not unusual for a party...