Moving between rooms

I have two Apple TVs in different rooms. I often move from one to the other and would love a simple way (ideally with Siri / shortcuts) to stop what I’m watching on one TV and have it start immediately (or be ready to just press / say “play”) on the other one.

That seems like it should be obvious, given that they’re set up with the same Apple ID on the same network with the same apps and the same Home Screen layout, but I don’t think there’s a way to do this.

Does anyone know if I’m wrong?

Thank you in advance :-)