Legislation that targets currency manipulation might make or break U.S. President Barack Obama’s signature Pacific trade deal. But calling out offenders for currency transgressions is far from a clear-cut exercise.

U.S. lawmakers, companies and unions are trying to use pending trade legislation to strike back at countries they say subsidize their industries through devalued exchange rates. They’re finding themselves entangled in a decadeslong debate about currency valuations that has bedeviled leading institutions such as...