DENVER—Mario Nicolais is the kind of candidate who could help the Republican Party win back the suburban voters it has lost. He's fiscally conservative but socially moderate, a member of the NRA but also a founder of a Republican group that champions gay rights. And on this afternoon at a local bar, he's clearly at ease chatting up the fans who've turned out to watch Mexico play Brazil in the World Cup. Hispanics make up 23 percent of his state Senate district, and Nicolais knows that Republicans will need to peel off some of them to have any hope of...