Rep. Eliot Engel and President Barack Obama didn’t always see eye-to-eye on issues of foreign policy. The New York congressman, as staunch a Middle East hawk as there currently is in the Democratic Party, was the most high-profile House Democrat to oppose Obama’s nuclear deal with Iran, the biggest foreign policy initiative of his presidency. Now, Engel, the chairman of the House Foreign Affairs Committee, is fighting for his political life amid a primary challenge to his left from Jamaal Bowman, a former high school principal. Obama administration alumni want him to know they haven’t forgotten his vote—and that they don’t especially like what he’s gotten done since. As some key figures in the party establishment, from Hillary Clinton to Speaker Nancy Pelosi to New York Gov. Andrew Cuomo, have lined up to support Engel, high-profile former Obama advisers, some of whom have immense sway with liberals nationwide through the popular podcasts from Crooked Media, have joined forces...