ROSWELL, Ga. — In the northern suburbs of Atlanta, where what is likely to be the most expensive House campaign in history is being waged, a band of conservative advocacy groups is grappling with a question that may decide whether the Republican Party keeps its House majority after 2018: Do you run with President Trump or against him?

Somehow, the groups are discovering, they will have to do both.

The race between the Republican Karen Handel and the Democrat Jon Ossoff in this reliably Republican district has become a proving ground for the surrogates tasked with defending the party’s majority. Both sides together could spend more than $40 million on the race to fill a seat that had been held by Tom Price, now the secretary of health and human services.

So far, Republicans have chalked up single-digit victories in Montana and Kansas special elections to replace other House Republicans tapped by Mr. Trump for his cabinet. Democrats made gains in both races from their 2016 showings. But for Republicans, a win is a win.

Georgia’s Sixth Congressional District, however, is different: an affluent, educated cluster of suburbs that barely went for Mr. Trump in November and could foretell the hazards in other suburban battleground contests in 2018.