What version of Mattermost are you using? And what version of MySQL? We’ve recently dropped support for MySQL 5.7, and will be deprecating support for MySQL entirely soon. Take a look at Mattermost changelog — Mattermost documentation and Migration guidelines from MySQL to PostgreSQL — Mattermost documentation
Can you log in to your database with the mysql command and the credentials / destination in DataSource from the Mattermost server?
What address does chatdatabase1.c5uemsg8swm9.us-east-2.rds.amazonaws.com resolve to for you? The error suggests that there’s a Layer 3 issue. What does your routing table look like?
Also, look at security group settings / firewall for the database instance.