|You could install the spare router you have at the TV instead of the switch but this would require extra setup on that router. To do this you would either need to disable the "router" part of the software, or you may have to get involved with static IP address and changing default gateways to point to your other router.|
If it's a router with 4 LAN ports, you would plug the uplink from the other router into a LAN port, and the clients into the LAN port.
The only real setup required would be to give the LAN side of that router an IP in the same subnet as the upstream router (preferably one outside of the DHCP scope) the same subnet mask, and make the gateway IP the IP of the upstream router's LAN side as you already pointed out. You would also want to ensure DHCP is shut off on the downstream router.
For me the above would take about 2 minutes. For someone who has less experience, 5 to 10 min's maybe. I only mention it because doing it this way would save you the cost of a switch, and, you wouldn't have a second router sitting around doing nothing but collecting dust.
Click on my name above in my response read my "how-to" guide on adding a second router for more info.