There are no issues connecting to it from any other environment (local works fine, as does from the Cloud), and the security group is currently totally open. An actual PostgreSQL instance, hosted in AWS, had no issue. It's just an Aurora thing. I've created several instances with different configurations. SSL authentication is turned off, and SSL itself is configured to be optional.
"Sorry, we encountered an error and were unable to complete your request."
It turns out that LookerStudio/Google's PostgreSQL connector is incompatible. I was able to connect using a third-party connector; SyncWith's (further down in the list), I think.
EDIT: It actually looks like, at the time that I tried to connect it, there might've been no tables or databases, or some other infrastructure issue, whereas the third-party connector didn't have that problem. It seems to be connecting with the native connector now just fine.