Kiwitrees should be able to work OK with either port 80 or 443, (see kiwitrees/includes/session.php lines 180-203) so there might be something else unusual at the server.
What does your phpinfo listing tell you about https settings (Administration > Site administration > Server information)?
Look particularly at these settings:
$_SERVER[‘REQUEST_SCHEME’]
$_SERVER[‘HTTP_REFERER’]
$_SERVER[‘HTTP_CF_VISITOR’]
$_SERVER[‘HTTP_X_FORWARDED_PROTO’]
$_SERVER[‘HTTPS’]