You should certainly have access on the basis of your oneworld international long haul flight LAX-SJO. "a oneworld international long haul flight is defined as an international flight marketed and operated by any oneworld carrier with a scheduled flight time longer than 5 hours". Source: https://www.oneworld.com/airport-lounge-results. Or more simply, AA explicitly says oneworld flights to Central America are eligible for access to their lounges.

Based on the 5h rule, I think this also means you should have access to the QF F lounge at LAX on the basis of your OWE status.