Title changed programmatically is not put into breadcrumbs
drupal_get_title() gets the title from a PHP static variable, which is different from the variable $variables['title'] passed to the template files. 

Your template hook would work if it uses the following code.

function lwt_preprocess_page(&amp;$vars) {
  $path = drupal_get_path_alias();
  if ($path == 'scores/completion') {
    // -1-
    $vars['title'] = t('Completion');
    drupal_set_title($vars['title']);
    // -1-
  }
}


What happens is the following:


lwt_preprocess_page() is invoked, and (through drupal_set_title()) sets the static variable containing the value returned from drupal_get_title().
template_process_page() is invoked, and it calls the theme function for the breadcrumbs; in your case that theme function is lwt_breadcrumb(), which calls drupal_get_title(), obtaining the same value you set in lwt_preprocess_page().


In lwt_preprocess_page() you could also replace the code between // -1- with the following one.

    unset($vars['title']);
    drupal_set_title(t('Completion'));


The reason also this code works is that template_process_page() first checks if $variables['title'] is set; if it is not set, it calls drupal_get_title().

  if (!isset($variables['title'])) {
    $variables['title'] = drupal_get_title();
  }


References


theme(), for the order in which theme template hooks are executed
drupal_static(), for understanding how the page title is stored in a static variable accessible to any function