With the method add_order_note you can add an order note to you order. If you look into the code of this method, you will see that this method is only a wrapper around the wp_insert_comment function from WordPress.
So your problem is, that you want to change the date of the order note. But that is not possible with the method add_order_note. If you look back into the code of that method you will see, that the method has only the
arguments “$note”, “$is_customer_note” and “$added_by_user”. So no date.
The easiest way would be to save the order note not with “add_order_note” but with the function “wp_insert_comment”.
$test_date="2005-08-05 10:41:13";
$note = __("Custom Order Note Here");
$user = get_user_by( 'id', get_current_user_id() );
$data = array(
'comment_post_ID' => $order_id,
'comment_author' => $user->display_name,
'comment_author_email' => $user->user_email,
'comment_author_url' => '',
'comment_content' => $note,
'comment_agent' => 'WooCommerce',
'comment_type' => 'order_note',
'comment_parent' => 0,
'comment_approved' => 1,
'comment_date' => $test_date,
);
wp_insert_comment($data);