My users are logged into the WP admin area. I modified the plugin code slightly to add the Intercom script to the WP admin footer:
add_action( 'wp_footer', array( $this, 'output_install_code' ) );
add_action( 'admin_footer', array( $this, 'output_install_code' ) );
It would be great if this were an option on the plugins options page so the plugin could be updated with overwriting the change.