Connect to RDS Private Instance from Docker Container
As per the error, it looks like the resolution is not happening for host xxxxx-xxxxx-xxxx.xxxxxxxxx.us-east-2.rds.amazonaws.com'. I mean the hostname is not getting resolved to the IP address from inside the container. 

You may need to check if the DNS resolver is working properly from inside the container or not. For testing, you can try to use the IP instead of the RDS endpoint hostname and see if it works.