Given that "press the case" is an idiom, any literal translation will not communicate in Spanish, unless it clearly duplicates the concept of the original.
https://www.macmillandictionary.com/...erican/press_2
to try to make someone accept something such as an opinion or a claim
press a point: She gave no answer, so I didn’t press the point.
press a claim: The young Duke of Normandy continued to press his claim to the English throne.
press a case: He was in Washington today to press the case for reforming tax laws.
... intentando convencer o persuadir (a alguien) sobre su opinión..., might do the trick.
Context is needed, of course, but that would be an option.