How to copy a PostgreSQL database at a filesystem level between servers
This is the script I ended up having to write to transfer all the files, meant to be run from the destination server.

#!/bin/bash
set -i

# Ensure databases on both servers are stopped.
sudo service postgresql stop
ssh -t myuser@oldserver "sudo service postgresql stop"

# Ensure our user owns postgres's files so our rsync call can access them.
ssh -t myuser@oldserver "sudo chown -R myuser:myuser /etc/postgresql"
ssh -t myuser@oldserver "sudo chown -R myuser:myuser /var/lib/postgresql"

# Copy down all the files.
sudo rsync --progress -azv --delete myuser@oldserver:/etc/postgresql/9.5/main/ /etc/postgresql/9.5/main
sudo rsync --progress -azv --delete myuser@oldserver:/var/lib/postgresql/9.5/main/ /var/lib/postgresql/9.5/main

# Fix permissions.
ssh -t myuser@oldserver "sudo chown -R postgres:postgres /etc/postgresql"
ssh -t myuser@oldserver "sudo chown -R postgres:postgres /var/lib/postgresql"
sudo chown -R postgres:postgres /etc/postgresql
sudo chown -R postgres:postgres /var/lib/postgresql

# Fix a missing ssl cert that pg uses for connections.
cd /etc/ssl/certs
sudo make-ssl-cert generate-default-snakeoil --force-overwrite

# Bring the server back up.
sudo service postgresql start


Rsync reported some errors, but I'm browsing my databases via psql and pgadmin3 and so far everything looks correct.

The fix for the SSL error I found here.