Requiring wp-login.php sends headers automatically

The login page is a horrible mess mixing functions with output the way it does. It makes projects like your unnecessarily difficult. It is actually better now that is used to be.

You can cheat around the problem with output buffering:

ob_start();
include('wp-login.php');
ob_end_clean();

Of course, the page still gets generated so it is consuming resources and not especially efficient, but you do get access to those functions. The downside– potentially a huge downside– is that that page does what it does in the background anyway. That could cause trouble if you aren’t careful.