If your DVD player is trying to access your NAS by its hostname, your Pi-hole might not be aware of that name, unless you've configured it for Conditional Forwarding or it would be acting as DHCP server.
I somehow doubt it would be a blocked public domain that is causing issues, but you also could use Pi-hole's Query Log to find out which of your DVD player's requests would be blocked.
The following FAQ might also be helpful in this regard.