A top National Labor Relations Board appointee of U.S. President Donald Trump should have recused himself instead of joining a Republican majority that limited which companies are subject to employee workplace claims, the agency’s inspector general said.

In a memo obtained by Bloomberg News, NLRB Inspector General David Berry flagged what he called “a serious and flagrant problem and/or deficiency” in the handling of conflict-of-interest issues by the agency. Berry’s memo concerned a December ruling in which the NLRB reversed...