HTML inside Customizer

I’ve had a look inside of the theme you’ve mentioned that does this. It seems like the code below (inside the zerif_customizer.js) file is what is making this button.

jQuery('.preview-notice').append('<a class="zerif-upgrade-to-pro-button" href="http://themeisle.com/themes/zerif-pro-one-page-wordpress-theme/" class="button" target="_blank">{pro}</a>'.replace('{pro}',zerifLiteCustomizerObject.pro));

This can be found in the themes js file. They’ve then hooked this file onto the customizer in the functions.php file using this code

function zerif_registers() {

    wp_enqueue_script( 'zerif_customizer_script', get_template_directory_uri() . '/js/zerif_customizer.js', array("jquery"), '20120206', true  );

    wp_localize_script( 'zerif_customizer_script', 'zerifLiteCustomizerObject', array(

        'documentation' => __( 'View Documentation', 'zerif-lite' ),
        'pro' => __('View PRO version','zerif-lite')

    ) );
}
add_action( 'customize_controls_enqueue_scripts', 'zerif_registers' );

Hopefully this gives you something to go on.