I've had my PS4 hooked up to the TV and streamed from the playstation TO the PC where the Hue software was running. Worked fine Hue wise, handling PS4 accounts was a bit tedious though.
Can you install an app called Hue Stream on the Shield? I use it on my Mi Box to sync the lights to what I'm watching. Works for most things.
Hey I can test it for you if you really want to know but in theory, that will work. I have play bars, the Sync Box, an OG NVidia Shield with GameStream, and a gaming PC with the sync software in the same room as the shield that runs a 2070 Super GFX so let me know if that set up works for your question. Here is my setup with the Sync Box and the Shield, you can see my gaming PC monitor the right of the TV, each of which has a playbar behind it. Should have low local latency as both the shield and the PC are wired connections. https://www.youtube.com/watch?v=3FmW0qXLJbE