For Google's sake, found out what was happening - it was simply my webhost denying outbound connections from the shared hosting. Doh! Double check with your webhost to ensure they're not doing sneaky things, people!
As a side note, I did finally get a connection working from a server on Amazon's own EC2 platform - however, the database connection string to PDO would NOT work with the port number on the end of the string.
So, if you're still getting errors, it's possible that Amazon's just telling you the wrong end point, and that you don't need to specify the host. All working now!