Hi!
That constant is defined in the ajax-search-pro\includes\classes\core\class-asp-manager.php file within this function:
As you can see the upload directory is requested through wp_upload_dir() call, which refers _wp_uploads_dir(), and should always return the correct upload directory.
Is this not the correct way? Please let me know, I am not too familiar with these modifications myself.
Also, from what I can see in these docs, if the UPLOADS constant is set in the wp-config.php file, the wp_upload_dir() will return that url relative to the ABSPATH constant. You might need to set that constant as well, I am not sure.
Let me know!