Log an error at shutdown priority 11 to confirm that is firing.

functions.php

<?php

add_action( 'transition_post_status', 'bk_shutdown', 11);
function bk_shutdown() {
 add_action('shutdown', 'bk_let_the_record_show', 11); 
}

function bk_let_the_record_show(){
  error_log( 'Shutdown at 11');
}