How to display dynamic content in start_lvl function

Finally, after some thorough search and as suggested by @Howdy_McGee in his comment I got the Nav Walker working as expected. For those looking for something similar, here is how I did it.

Instead of outputting parent menu item title in start_lvl function, I included it in start_el function and the final code looked like following:

class my_custom_navwalker extends Walker_Nav_Menu {

    function display_element($element, &$children_elements, $max_depth, $depth = 0, $args, &$output)
    {
        $id_field = $this->db_fields['id'];
        if (is_object($args[0])) {
            $args[0]->has_children = !empty($children_elements[$element->$id_field]);
        }
        return parent::display_element($element, $children_elements, $max_depth, $depth, $args, $output);
    }

    function start_lvl(&$output, $depth = 0, $args = array()) {

        $indent = str_repeat("\t", $depth);

        if ($depth == 0) {

            $out_div = '';
        }
        else {
            $out_div = '';
        }
        // build html
        $output.= "\n" . $indent . $out_div . '<ul>' . "\n";
    }

    function start_el(&$output, $item, $depth = 0, $args = array() , $id = 0) {
        global $wp_query;

        $indent = ($depth) ? str_repeat("\t", $depth) : '';

        $class_names = $value="";
        $classes = empty($item->classes) ? array() : (array)$item->classes;
        $class_names = join(' ', apply_filters('nav_menu_css_class', array_filter($classes) , $item));
        // $class_names=" class="". esc_attr( $class_names ) . '"';

        if ($args->has_children && $depth == 0) {
            $has_sub = ' has-sub';
        }

        $output.= $indent . '<li id="menu-item-' . $item->ID . '"' . $value . 'class="' . $class_names . $has_sub . '">';

        $attributes = !empty($item->attr_title) ? ' title="' . esc_attr($item->attr_title) . '"' : '';
        $attributes.= !empty($item->target) ? ' target="' . esc_attr($item->target) . '"' : '';
        $attributes.= !empty($item->xfn) ? ' rel="' . esc_attr($item->xfn) . '"' : '';
        $attributes.= !empty($item->url) ? ' href="' . esc_attr($item->url) . '"' : '';
        $prepend = '';
        $append = '';

        //$description = !empty($item->description) ? '<div class="">' . $item->description . '</div>' : '';
        // if($depth != 0)
        // {
        //         $description = $append = $prepend = "";
        // }


        $item_output = $args->before;

        $item_output.= '<a' . $attributes . '>';

        $item_output.= $args->link_before . $prepend . apply_filters('the_title', $item->title, $item->ID) . $append;
        $item_output.= '</a>';
        //$item_output.= $description . $args->link_after;


        if ( $args->has_children && $depth == 0) {

            $item_output .= ' <div class="dropdown-wrapper"><div class="dropdown"><div class="item-title-class">'.$item->title.'</div>';

        }

        $item_output.= $args->after;
        $output.= apply_filters('walker_nav_menu_start_el', $item_output, $item, $depth, $args, $id);
    }

    function end_lvl(&$output, $depth = 0, $args = array())
    {
        $indent = str_repeat("\t", $depth);
        if ($depth == 0) {
            $out_div_close="";
        }
        else {
            $out_div_close="";
        }
        $output.= "$indent" . "\n";

        $output.= "</ul>" . $out_div_close . "\n";
    }
}