WpEngine has a too restrictive security system. We had many problems with WpEngine related to Backup, but haven’t found the exact source of the problem. We contacted WpEngine tech supported and even they didn’t understood why our Backup is not working.
They have set restrictions on the production version of the site, if you want to install Demo content you need to create a staging (sandbox) project, install demo content in staging mode, and deploy to production when demo content installed Staging site.
For more info check this video screencast