The Iran war is going badly for the United States, with no off-ramp in sight. Still, other than President Donald Trump, few have continued to champion the U.S. military campaign as enthusiastically as Defense Secretary Pete Hegseth, who touts lethal American firepower, critics say, with something akin to a schoolboy’s excitement about the grim business of war.

Now, as next week’s midterm elections approach, Mr. Trump’s support for his defense secretary is being tested: Promises of a quick victory against the Islamic Republic have given way to high gas prices, escalating ultimatums against an uncowed regime, and increasingly sharp questions from Congress over strategy and war costs.

If Republicans lose control of Congress next week, Mr. Hegseth could be the logical scapegoat, analysts say. Polls already rank the defense secretary high among administration officials with particularly low approval ratings.…