WEST DES MOINES, Iowa—On a recent ice-swept morning, a group of self-described moderate Republicans met in a hotel convention room, looking to find a way to break the chill of a presidential season that has found many of them left out in the cold. "Our goal is to get traditional centrist moderate Republicans to get to the caucus and make their voices heard," said former Iowa Lt. Gov. Joy Corning. "The moderates who are out there, they've been rather quiet for a few years. Many of them have dropped out of the party or become independents, and so this is...