Amazon DMS (database migration service) actually works really well and allows you to do this without writing any code. You may need to change your Postgres conf

Amazon DMS (database migration service) actually works really well and allows you to do this without writing any code. You may need to change your Postgres config a bit. DMS isn't just for moving two RDS it can also be used to move off or RDS or even move from one non-RDS server to another non-RDS server that isn't even on AWS. It can also do Postres to MySQL, MySQL to Postgres, etc. It also does continuous replication so once the copy is done it will keep up to date (yes, even if it is Postgres -> mySQL). One gotcha, though, it does not bring over secondary keys so make sure to recreate them before sending traffic over. But this article does provide some insight on how to configure Postgres to be similar to RDS in terms of functionality.