WordPress is designed so you should never need to have duplicate installs, even if you have thousands of sites across hundreds of domains.
With that in mind, I’m curious why, if you’re just making one page different, you don’t use a custom page template instead of the bloat of multiple installs? It seems it would be much, much easier – just make a copy of page.php, rename it page-coolpage.php, and modify it to do exactly what you need. WordPress is designed with that in mind, so it makes more sense to make use of the tools built in that throwing in a duplicate install.
Alternately, make your WordPress a multisite install, deploy the second theme on the coolpage site, and go from there. Single dashboard = one place for updates, comment management, content management, and so on. It’s a little more complex than a custom page template, but it’s less overhead and hassle than multiple installations of WordPress.
As someone who is cleaning up the mess left behind by someone doing exactly what you’re thinking of doing (multiple installs), I assure you a custom page template or multisite setup is infinitely preferable to managing multiple installations.