I have ATT/Uverse for TV and internet as well. The outside wiring (I believe this is a four wire line) connects to the ATT gateway (modem). This then outputs the phone signal into a phone jack, has four Ethernet ports on the back, and also has a wireless router. I don’t have any coax involved.
To move the gateway, you would have to reroute the outside wiring to the new location, reroute any Ethernet cables to the new location, and have a phone jack available to plug it into. Assuming you have a phone jack available, that will be as simple as unplugging from the jack in the spare bedroom, and plugging it in to one in the living room. You can then connect the gateway and TV unit using an Ethernet cable. If the TV and phone jack are not near each other, you’ll have to have either a phone line or an Ethernet cable running from the TV location to the phone jack location, depending on where you put the modem.
Your main issue will be moving the wiring from the outside so that it comes into the house near your TV. In my case, they ran it along outside the house, and came in the room where we wanted the gateway. If they did that for you, you’d have some wire to work with, but if there’s not enough length to get to the new location, you’d need to splice in more line. It could be important to do that splice well, I’m not sure. In any case, it’s not just a matter of a splitter. On the other hand, if there’s enough line, it should just be a matter of rerouting it to the Living room. You need to find that line, and decide if you have enough length to just move it.
Another approach which might be easier for you would be to connect a router to the Ethernet line at the TV, and plug the TV into that. I have a new non-wifi router at the TV, and the TV (and a PS3) hooked into that, and that works well.
That might not work with an older Wifi router, though. I initially tried it with our older Wifi router, and it couldn’t handle the traffic the TV needed. I instead connected the older Wifi router to that, to have WiFi in that room, although I don’t need the second Wifi router any more.