WASHINGTON—As one of President Trump’s top compliance and ethics attorneys in the White House, Uttam Dhillon had urged several candidates for Drug Enforcement Administration chief to withdraw from consideration, citing concerns about their background checks. Then, he accepted the job himself.

Mr. Dhillon’s rise to the top of the world’s largest drug-fighting agency—after being closely involved in the selection process—has riled police groups that had pushed the White House to choose a DEA administrator with a law-enforcement...