You can't do that with a Chromecast. The goal here is to sell devices which can ensure they always talk to their own DoH endpoint with no ability to secure it.
Blocking people from using DoH isn't going to solve the problem you are describing, it will just weaken users' privacy. Big vendors like Google will just create their own workarounds.
Before DoH though, my understanding is a good gateway appliance could just overwrite those DNS requests as it sees them. Now they'll be encrypted, and you'll have no way to tell the Chromecast where to connect (or even to where it is connecting).
All the major appliance vendors (Google, Amazon) already have huge fixed IP ranges to devote to this purpose, which are effectively unblockable because they might be shared with important cloud services.