Normally.. from MY experience.. IF there is any setting on a device for the DNS, it will always take presidence over anything in the gateway.
Normally.. if you have a device set up via DHCP.. it will get an IP address, the gateway address... and then often either specifically assigned DNS thats stored in the gateway, or will just take the gateways IP address as the DNS, and the gateway does forwarding of the DNS through what it has stored.
But.. if you have a DNS specified on the Device itself.. it will use that. (i have my PCs set up this way).
I am guessing that you are trying to use a 3rd party DNS to use US netflix, etc?
Unfortunately, from what i can find.. there is only one way to do this.
You need to use a 3rd party router, using something like DD-WRT on it... which allows some pre-routing commands done, to change where those google DNS requests go to.
Alternatively, it looks like you can ROOT your chromecast, to a version which allows DNS changes as well