Programmatically trigger AJAX and return HTML from menu

/**
 * Implements hook_menu().
 */
function trigger_ajax_menu() {
  return array(
    'custom-ajax/%' => array(
      'page callback' => 'trigger_ajax_update',
      'page arguments' => array(1),
      'access callback' => TRUE,
      'delivery callback' => 'ajax_deliver',
      'theme callback' => 'ajax_base_page_theme',
      'type' => MENU_CALLBACK,
    ),
  );
}
 
function trigger_ajax_update($uid) {
  $account = user_load($uid);
  $content = '<div id="target-element-update">Hello ' . $account->name . '!</div>';
  return array(
    '#type' => 'ajax',
    '#commands' => array(
      ajax_command_replace('#target-element-update', $content)
    ),
  );
}

(function ($) {
  $(function () {
    /**
     * Create a placeholder function that we can execute to trigger an AJAX call.
     */
    Drupal.ajax.prototype.triggerAjax = function (uid) {
      // Leave if we're already executing.
      if (this.ajaxing) {
        return false;
      }
 
      // Append the User ID to the end of the URL to complete our menu entry.
      this.options.url += uid;
 
      $.ajax(this.options);
    };
 
    Drupal.ajax['triggerObjectPlaceholder'] = new Drupal.ajax(
      null,
      $(document.body),
      {
        url: '/custom-ajax/',
        event: 'onload',
        keypress: false,
        prevent: false
      });
 
    // Some event
    // ...
    // Trigger the manual ajax trigger
    var userId = 1;
    Drupal.ajax['triggerObjectPlaceholder'].triggerAjax(userId);
  });
})(jQuery);