All of the following refers solely to the third environment (WPEngine production site using their custom caching system).
Further testing shows that the Cookie is being properly set and sent by my plugin. I did a WP_DIE() right after setting the cookie, and the Cookie was present in my browser. With the WP_DIE() removed, so that WordPress can complete processing for the page, the Cookie does not appear in my browser.
I also proved, by way of an alert() in JavaScript, that the JavaScript is always being run when it should be.
After staring at the list of plugins that are activated, I was about to give up. Less than 5 minutes later, as I started to do something else, I suddenly thought of the Membership plugin.
I deactivated the plugin Membership System v2 and everything works. I activated it and it doesn't work, i.e. - the cookie does not appear in my browser.
At this point, I turn this over to you. Quite frankly, I am out of ideas. Membership System v2 is a very expensive paid plugin that requires IOCube Loader to run.