Optimal RBR50 router, modem, switch, satellite setup

I have an RBR50 router with 2 satellites. For the past 6 years I have had the router's Internet port connected to my home's switch, and the switch connected to my fios modem. I did this because the switch and modem are connected to a battery backup, so if the power goes out, at least I still have wired internet. This setup has worked relatively fine for the past 6 years, except that there are devices that connect to the router that would be better off connecting to a satellite. The router is in AP mode.

Is there anything wrong with my wiring? Would I be better off connecting the router's Internet port to my modem, and then running another cable from one of the router's Ethernet ports to my home's switch? What would be the benefit?