The woman tasked with turning around Donald Trump’s Republican presidential campaign has toiled extensively in the party’s antiestablishment conservative lane during the past decade, working for House members who tried to overthrow former Speaker John Boehner and Senate candidates whose stumbles helped delay the takeover of that chamber.

Kellyanne Conway, a longtime Washington-based pollster, was installed this week as manager for the New York billionaire’s insurgent campaign. A New Jersey native with a knack for snappy sound...