To win re-election, Mr. King must first defeat three primary challengers, most prominently Randy Feenstra, a state senator who has outraised him with the support of the Republican establishment. Mr. King brought in just $91,000 in the quarter ending in June, compared with Mr. Feenstra’s $140,000.

Money, however, has never mattered much in Mr. King’s re-elections. He has nearly universal name recognition in his district, and, until recently, voters broadly embraced his pugnacious personality and positions on bedrock conservative issues like abortion and gun rights.

Mr. Scholten, who outspent Mr. King nearly four to one in 2018, came within three percentage points of unseating him after the congressman endorsed a candidate for Toronto mayor with neo-Nazi ties and, in an interview with an Austrian publication that surfaced late in the race, seemed to endorse the “Great Replacement,” a conspiracy theory espoused by white supremacists.