WASHINGTON—In a private room deep inside the U.S. Capitol complex, Rep. Gerry Connolly fumed as he questioned a former Trump administration official over the security-clearance process.

It wasn’t the official’s answers that irritated the Virginia Democrat but what he perceived as the constant objections and an “arrogant” tone from White House attorney Mike Purpura. “Be respectful of the branch you’re in front of, and if you don’t want to do that, we can subpoena your witness and have that clash,” Mr. Connolly recalled telling...