When you open Add Device, what do you see as the addresses of the DirecTV device? Then, when you look on the Sharp TV me — Staff replied: “We would suggest emailing your configuration to support so that they can take a look at it.”
When you open Add Device, what do you see as the addresses of the DirecTV device? Then, when you look on the Sharp TV menu for the IP Address, what does that say? Finally, what is the Wi-Fi IP Address of your iOS device? It sounds like there is some kind of proxying going on in your router between your wireless and wired network and looking at those 3 addresses should prove or disprove that. Assuming that's the case, you'd need to turn that off.