XML Sitemap Not Parsing

I've set up an XML sitemap using this navigation template (replacing example URL with my own):

<perch:before><?xml version="1.0" encoding="UTF-8"?>
<urlset xmlns="https://www.sitemaps.org/schemas/sitemap/0.9">
</perch:before>
<url>
    <loc>https://example.com<perch:pages id="pagePath" /></loc>
</url>
<perch:after>
</urlset>
</perch:after>

And a file in the root of the website called 'sitemap.xml' which contains this (my Perch directory is the default '/perch'):

<?php
    header('Content-type: application/xml');
    include('perch/runtime.php');
    perch_pages_navigation(array(
        'template' => 'sitemap.html',
        'flat' => true,
        'hide-extensions' => true
    ));
?>

I've then added this line to my .htaccess file to parse the XML as PHP:

AddType application/x-httpd-php .php .htm .html .xml

Now, I can't view the XML file (it just downloads a copy when I browse to the file) and the Google Sitemap test is throwing this error: "Your Sitemap appears to be an HTML page. Please use a supported sitemap format instead."

Perch doesn't appear to be doing anything to the file - is this pointing to a problem in my .htaccess file?

My hosting is with Media Temple which uses the same MIME Type and as far as I can see, it should be working.