Imran Awan was arrested at Dulles International Airport July 24, while attempting to board a flight to Pakistan. For more than a decade the congressional staffer had worked under top House Democrats, and he had just been accused by the FBI of bank fraud.

It was a dramatic moment in a saga that started in February, when Capitol Police confirmed an investigation into Mr. Awan and his family on separate accusations of government theft. The details are tantalizing: The family all worked for top Democrats, were paid huge sums,...