This is probably a ridiculous question, but I usually stream from my laptop to my LG tv or my phone to any other tv, and I find that using my VPN keeps the casting output option from working. Like in Popcorntime, the Watch Now doesn't show my tv, only the laptop app or VLC. On my daughter's fire stick I can't even cast to it unless I turn my VPN off.
I don't have any problem streaming Plex or Emby from my laptop to the tv. Is there a way to cast without compromising my privacy?
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.