If you take a look at the source code of this part …
do_action( 'customize_controls_print_styles' );
do_action( 'customize_controls_print_scripts' );
?>
</head>
<body class="<?php echo esc_attr( $body_class ); ?>">
<div class="wp-full-overlay expanded">
<form id="customize-controls" class="wrap wp-full-overlay-sidebar">
<div id="customize-header-actions" class="wp-full-overlay-header">
<?php
$save_text = $wp_customize->is_theme_active() ? __( 'Save & Publish' ) : __( 'Save & Activate' );
submit_button( $save_text, 'primary save', 'save', false );
?>
<span class="spinner"></span>
<a class="back button" href="https://wordpress.stackexchange.com/questions/96947/<?php echo esc_url( $return ? $return : admin_url("themes.php' ) ); ?>">
<?php _e( 'Cancel' ); ?>
</a>
</div>
… you can see there is no action for this place.
But you can hook into customize_controls_print_scripts
and register a new JavaScript that inserts a button.
add_action( 'customize_controls_print_scripts', 'add_customizer_button' );
function add_customizer_button()
{
wp_register_script( 'my_customizer_script', plugins_url('/customizer_button.js', __FILE__) );
wp_enqueue_script( 'my_customizer_script' );
}