I thought I would provide an update as I have made my own patch to do this.
Essentially I had to configure Moodle via Home / Site administration / Security / HTTP security / Allow frame embedding and set it to enabled.
Additionally, I had to edit httpd.conf in Apache and adjust the configuration options related to 'X-Frame-Options' so that they do not default to the "sameorigin" settings.
Making the above changes and restarting Apache allowed an external site IFrame to load any page from the Moodle instance.
The next step was to create a PHP file that would display only the badges. I duplicated the badges/badge.php file and modified it by removing the header/footer output statements. I also duplicated and modified the respective rendering classes in badges/renderer.php so that only the badge icons were displayed.
Given that I had already done the single sign-on between the external site system and Moodle, the IFrame had access to the Moodle session and allowed seamless log-in + access to any Moodle pages.
Completing the above steps resulted in a patch that now enables display of Moodle badges within the external site IFrame. This should be extensible to any aspect of Moodle e.g. displaying of enrolled course names as long as the corresponding Moodle php has been created.
Hope this helps anyone with a similar use-case requirement. I would be glad to help if you have any queries on this.