@PeterT
Thank you for making me aware of R2 Reader (for iOS 

 )! 
Using it I can confirm your problem.
If the calibre library is outside of the COPS path (for me: /var/www/html/cops/ ) R2 Reader does not show the download button. 
Adding     
$config['cops_use_url_rewriting'] = '1';  to config.local.php fixed the problem for me.
Let us know if this will fix your problem as well.