How can I retrieve the list of all databases in an RDS DB instance via API?

It's not possible because each database engine has a different implementation of the database concept, so it's a subject that varies from engine to engine.

What you can do is to use the ExecuteStatement RDS API command https://docs.aws.amazon.com/rdsdataservice/latest/APIReference/API_ExecuteStatement.html to run a select that queries the DB data dictionary for that specific engine for that RDS instance, and then you can get the database names.

It does not avoid to run the SQL, but it is a standardized way to get this info no matters the instance DB engine, as the result output has always the same structure.