Maybe I’m wrong but for the sake of trying to get you started down the right path until someone smarter than me answers for sure. I think you would need to configure a split tunnel on your device. Even then I don’t think that would work either. I think that “casting” just tells the display the server address and it tracks down the stream info. Your casting device then just functions as a remote through the network.
So even if It could work, you’re home network address would be compromised by the network traffic being sent to your display without a VPN.
Like another person has said, your best bet is more likely torrenting to Plex, Emby, Jellyfin then casting from your media server.