WordPress sending 404 when hooking into send_headers

I’m trying to hook into the send_headers function of WordPress so I can check to see if a sitemap.xml was requested. That way, I can serve up a custom PHP function to automatically generate the XML based on WordPress posts and pages.

add_action( 'send_headers', 'custom_send_headers');
if( !function_exists('custom_send_headers') ) :
    function custom_send_headers( )
    {
        global $route, $wp_query, $window_title;
        $bits = explode( "/", $_SERVER['REQUEST_URI'] );

        if( $bits[1] === 'sitemap.xml' )
        {
            if ( $wp_query->is_404 ) {
                $wp_query->is_404 = false;
            }
            include('sitemap.php');
            die();
        }
    }
endif;

Everything is serving up properly, and my PHP file is including the appropriate headers:

<?php header('Content-Type: application/xml'); ?>
<?php echo '<?xml version="1.0" encoding="UTF-8"?>'; ?>
<urlset xmlns="http://www.sitemaps.org/schemas/sitemap/0.9">
    ....
</urlset>

Why would WordPress be sending 404?

I tried using WordPress templates and that didn’t seem to work.

$template = locate_template('sitemap.php');
load_template($template);

Is there another function I should be hooking into? Did I miss something else that I should be doing?