I had so much trouble doing this - I thought I would make a Q/A on StackOverflow to explain the process.
The question is about copying an RDS postgres database for development usage - especially for testing database migrations scripts, etc. That's why the focus on a "single schema" within a "single database".
In my case, I want to create a test database that's as isolated as possible, while remaining within a single RDS instance (because spinning up entire RDS instances takes anywhere from 5 - 15 minutes and because I'm cheap).