How to include a third level with wp_get_nav_menu_items function

Seems that nobody has an answer to this question, so I followed @Zlatev advice and created a custom walker class…

Download the plugin from GIT and add it to your theme, add the custom walker below to a include() or directly into your functions.

<?php 
    /**
     * Required plugin
     * https://github.com/adgsm/multi-level-push-menu
     */
    class Push_Menu_Walker extends Walker_Nav_Menu {
        /**
         * Start the element output.
         *
         * @param  string $output Passed by reference. Used to append additional content.
         * @param  object $item   Menu item data object.
         * @param  int $depth     Depth of menu item. May be used for padding.
         * @param  array $args    Additional strings.
         * @return void
         */
        function start_el( &$output, $item, $depth = 5, $args = array(), $id = 0 ) {

            $output .= "<li>";

            $attributes="";
            $title = $item->title;
            $desc = $item->description;
            $classes = $item->classes;
                    /*["classes"] => array(8) {
                        [0]=> string(0) "" <-- THIS IS THE CUSTOM CLASS
                        [1]=> string(9) "menu-item" 
                        [2]=> string(24) "menu-item-type-post_type" 
                        [3]=> string(21) "menu-item-object-page" 
                        [4]=> string(17) "current-menu-item" 
                        [5]=> string(9) "page_item"
                        [6]=> string(12) "page-item-50"
                        [7]=> string(17) "current_page_item"
                    }*/
            $font_awesome_class=" class="fa fa-". $classes[0] . '"';

            ! empty( $item->url )
                and $attributes .= ' href="'   . esc_attr( $item->url        ) .'"';

            $title = apply_filters( 'the_title', $item->title, $item->ID );

            if (!empty ( $classes[0] )) : // If we have a custom class, add the H2 + icon
                    $item_output = $args->before
                        . "<a $attributes>"
                        .       $args->link_before
                        .       $title
                        . '</a> '
                        . '<h2>'
                        .       '<i ' . $font_awesome_class . '></i>'
                        .       $title
                        . '</h2>'
                        . $args->link_after
                        // . $description <-- Not needed for now...
                        . $args->after;
            else :
                    $item_output = $args->before
                        . "<a $attributes>"
                        . $args->link_before
                        . $title
                        . '</a> '
                        . $args->link_after
                        . $args->after;
            endif;

            // Since $output is called by reference we don't need to return anything.
            $output .= apply_filters(
                'walker_nav_menu_start_el',
                $item_output,
                $item,
                $depth,
                $args
            );
        }
    }
 ?>

Finally call the walker in your theme:

1. Update the theme_location
2. Remove the container by creating an empty argument
3. Call the walker we’ve just created

<div id="push-nav">
<nav>
<h2>Navigation <i class="fa fa-reorder"></i></h2>
<?php wp_nav_menu( array( 'theme_location' => 'menu-3', 'container'=>'', 'walker' => new Push_Menu_Walker()) ); ?>
</nav>
</div><!-- /#page -->