GRAND RAPIDS, Minn. — For 70 of the past 72 years, a Democrat has represented this rural corner of northern Minnesota in Congress. But when loggers, foresters and truckers convened on the county fairgrounds here for the annual timber industry expo last month, the star of the “celebrity log-loading” contest was a Republican.

He is Pete Stauber, a former professional hockey player and retired police lieutenant with a ramrod-straight bearing and a politician’s firm grip. He may also be his party’s best hope for pulling off this year’s most improbable feat: flipping a Democratic House seat.

As Republicans brace for a “blue wave” that could cost them control of Congress, they can count on one hand their opportunities to play offense. Mr. Stauber’s race — in a union-heavy, Trump-friendly, mostly white working-class district that includes the mining region known as the Iron Range — is their best shot.

Of the 70 House races considered competitive by the nonpartisan Cook Political Report, just one — Mr. Stauber’s — involves a Democratic seat that now leans Republican.