Force absolute URLs for CSS / Javascript in head?

I'm not sure I fully understand what the third party site is doing but one option may be to add/change the $base_url in /sites/default/settings.php to be the absolute path of your site.

Copied from settings.php:

/**
* Base URL (optional).
*
* If you are experiencing issues with different site domains,
* uncomment the Base URL statement below (remove the leading hash sign)
* and fill in the URL to your Drupal installation.
*
* You might also want to force users to use a given domain.
* See the .htaccess file for more information.
*
* Examples:
* $base_url = 'http://www.example.com';
* $base_url = 'http://www.example.com:8888';
* $base_url = 'http://www.example.com/drupal';
* $base_url = 'https://www.example.com:8888/drupal';
*
* It is not allowed to have a trailing slash; Drupal will add it
* for you.
*/
# $base_url = 'http://www.example.com'; // NO trailing slash!

I'm also looking for a fix here without hacking anything core.

I have the same problem

Check out the method used by the web_widget module to change relative path links to full url links

from web_widget.module

/**
 * Stores the fact that if we're inside web_widget processing.
 */
function web_widgets_inside_widget($in = NULL) {
  static $base_path;
  if ($in === TRUE) {
    $base_path = $GLOBALS['base_path'];
    $GLOBALS['base_path'] = url('', array('absolute' => TRUE, 'purl' => array('disabled' => TRUE)));
  }
  elseif ($in === FALSE) {
    $GLOBALS['base_path'] = $base_path;
  }
  static $inside = FALSE;
  if (is_null($in)) {
    return $inside;
  }
  else {
    $inside = $in;
  }
}

maybe you could use a similar approach for the pages requested by the third party processor