WASHINGTON — The conversation at a London bar in September 2016 took a strange turn when the woman sitting across from George Papadopoulos, a Trump campaign adviser, asked a direct question: Was the Trump campaign working with Russia? The woman had set up the meeting to discuss foreign policy issues. But she was actually a government investigator posing as a research assistant, according to people familiar with the operation. The F.B.I. sent her to London as part of the counterintelligence inquiry opened that summer to better understand the Trump campaign’s links to Russia. The American government’s affiliation with the woman,...