A contentious two-year dispute between one of the District of Columbia’s most prominent law firms and one of its top public affairs outfits has landed in court, offering a rare glimpse into the inner workings of a major public-influence campaign normally shrouded in secrecy.

APCO Associates has sued Akin, Gump, Strauss, Hauer & Feld for $1.3 million over allegations of fraud and breach of contract.