Donald Trump has said over the years that he welcomes and even encourages rivalries in his administration, and delights in watching aides compete to please him. But for the past year, the president has allowed a rift to widen within the team tasked with delivering on the mass-deportation plan that is his most important domestic-policy initiative. That has led to months of acrimony and left many veteran officials at DHS—including those who support the president’s deportation goals—astonished at the dysfunction.