How can I display Custom Post type Custom Columns and its Content in a Dashboard Widget?

To answer your specific question: no, there are no hooks that would allow you to “reuse”. However, there is nothing
keeping you from calling your browseCustomColumns() function in your function that renders the dashboard widget.
However, in order to do that, you’d have to make a minor change your compile_dashboard_widget() function:

public
function
compile_dashboard_widget ()
{
    // removed the call to echo, just call $this->dashboard_widget_html () and have
    // $this->dashboard_widget_html () do all the echo'ing
    $this->dashboard_widget_html () ;
    $this->loadPluginScripts () ;
    $this->loadFrontScripts () ;

    return ;
}

public
function
dashboard_widget_html ()
{
    global $post ;

    $args = array (
        'post_type' => $this->post_slug,
        // any other WP_Query args you want
        ) ;
    $posts = new WP_Query ($args) ;

    if ($posts->has_posts ()) {
        while ($posts->has_posts ()) {
            $posts->the_post () ;

            // you'd probably want addition markup here
            // (e.g., a table, which each post in a <tr> and title/type/cost as <td>)
            // but you can figure out how you want your dashboard widget to look
            the_title () ;

            $this->browseCustomColumns ('wp_type', $post->ID) ;
            $this->browseCustomColumns ('wp_cost', $post->ID) ;
            }
        }
    else {
        echo 'no custom posts' ;
        }

    wp_reset_postdata () ;

    return ;
}

Hope this helps.