You could create a dummy user (with no rights) and log in every visitor using this dummy user. This could be done pretty easily:
But in my opinion that's a rather flawy workaround.
Your main problem is that all pages for not-logged-in users come from the page-cache. That also means that all users get the same html output as long as there is page-caching active. So there can't be any difference to your pages based on the visitor. All visitors get the same output.
A better way would be to create a ticket at WPEngine and ask them to disable page-caching for your site. That's actually also the way they suggest on the page you linked to.