This is 100% not possible. The Chromecast always hits Google's DNS to resolve the media server's address. Which means you must host your media under a resolvable domain name, exposed to the internet.

This kind of thing (assuming it can resolve through Google DNS) would short-circuit that requirement. You could feed it a domain name that points back to a local address and not expose your server to the internet.

Of course there are other ways around the problem, but this would be far, far simpler. Honestly it seems like it'd be faster and simpler than setting up a subdomain and reverse proxy, which is what I did.