Last fall, federal agents raided the home and office of Robin L. Raphel in search of proof that she, a seasoned member of America’s diplomatic corps, was spying for Pakistan. But officials now say the spying investigation has all but fizzled, leaving the Justice Department to decide whether to prosecute Ms. Raphel for the far less serious charge of keeping classified information in her home. The fallout from the investigation has in the meantime seriously damaged Ms. Raphel’s reputation, built over decades in some of the world’s most volatile countries. If the Justice Department declines to file spying charges, as...