I'm doing some more testing but it appears that there's something going on with Docker. I don't know if it's Unraid or if it's something with the container and the way the networking layer interacts with the Nvidia Shield. But I moved my DVR over to a Windows Virtual Machine and suddenly things are a whole lot more stable. Slight bonus is that virtual machine has a NVIDIA RTX 4060 passed through to it, so nice for transcoding.
My Shield in my bedroom that's over Wi-Fi would drop connection after about 10 minutes of watching TV, maybe even sooner. And I just chalked it up to it being Wi-Fi. But that stopped as soon as I moved to Windows.
I'm going to need a few days or at least a few hours to test, but I think moving my server to my Windows Virtual Machine is probably the solution in the end. And there's something going on with the Docker container potentially or at the very least, the way the Docker runs on Unraid.