Решено! Вывод меню с помощью wp_nav_menu shortcode

Регистрируем новое меню WordPress

add_action('after_setup_theme', function(){
  register_nav_menus( array(
    'english_menu' => __( 'English menu', 'english_menu' ),
  ) );
});

Создаем wp_nav_menu shortcode

function eng_menu($atts, $content = null) {
	extract(shortcode_atts(array(  
		'menu' => 'English menu',
		'container'       => 'div', 
		'container_class' => 'gds-main-menu', 
		'menu_class' => 'nav-menu', 
		'echo'            => true,
		'theme_location'  => 'english_menu',), 
		$atts));
return wp_nav_menu( array( 
		'menu'            => $menu, 
		'container'       => $container, 
		'container_class' => $container_class, 
		'menu_class'      => $menu_class, 
		'echo'            => false,
		'theme_location'  => $theme_location));
}
add_shortcode("englishmenu", "eng_menu");