The early days of Ms. Haley’s campaign, which she announced on Tuesday, quickly illustrated the challenges facing Republican women. But for Republican women, that double bind comes with a twist. Even before she entered the race, Mr. Trump dismissed Ms. Haley as “overly ambitious,” which struck some observers as sexist. While Ms. Haley has joked about the pain her heels inflict on antagonists, Ms. Greene has mocked Ms. Haley and Ms. Cheney as the equivalent of a “Bush in heels” to deride them as moderates. “There’s this feeling, like, ‘Well, even if I’m ready to vote for a woman president, I’m not really sure everyone else is.’ And that holds women back.”The post Haley Walks Treacherous Road for G.O.P.