This error appears when navigating to Performance, or the main W3TC settings page.
This is because of a bug in the latest CloudFormation template on the AMI Marketplace page. An update is currently pending, but you can download the corrected template here.
To fix this issue, run the following commands:
cd /var/www/html/wordpress
chown -R nginx:nginx .
This sets the ownership of all docroot files to nginx.
Alternatively, you can run this command that fixes the permissions:
/home/ec2-user/wordpress-fix-permissions.sh