How do I call the Fatal Error Handler for extern/url.php?
When extern/url.php is invoked, the CMS is not bootstrapped and extern/url.php require_once's all the code it needs to run. So if you specify a Fatal Error Handler (CiviCRM > Administer > System settings > Debugging and Error Handling) it won't be found.

To resolve this, edit extern/url.php to add a require_once statement which names the file containing your handler. (Be careful this file doesn't execute code that will impact on extern/url.php - mine only has function definitions.)