You’re right, Felix. WP Engine does not allow server configuration access. That means this plugin will not work on a WP Engine account.
All hope is not lost, OP. As @4briang said, WPE is rolling out Let’s Encrypt access to all customers. You may have the ability to use it already.
To find out if you do, log into the WPE user portal, select your install, and then click the “SSL” link. From there, click the blue button labeled “Add SSL Certificates” and see if Let’s Encrypt is an option.
Hope this helps!