In that case, running a relay node on your router would be best, the thing which responds to DHCP requests on your LAN and can set itself as the default gateway. That way all packets from Android TV for 100.640.0/10 will go to the default gateway without needing to modify the routing table on the Android TV device.
If your router is something you can install software on, like OpenWRT or a Raspberry Pi, getting tailscaled running should work.
Also: if you’re not doing so already, starting the subnet relay with
--advertise-routes=... --snat-subnet-routes=false would likely work best. snat-subnet-routes tells it to just send the 100.64.0.0/10 addresses out onto the LAN, and allow the other devices on the LAN to see the actual IP address to send their packets back to.