First your needs are simple and a small inverter will work fine. But for an inverter, any size, you need adequate wiring from the battery to the inverter. I don't think the antenna wiring is going to be adequate.
I'd suggest running 12 ga. wire directly from your battery (fused) and grab the negative at the closest point to the inverter or directly to the battery also. OR: if the TV and components are on a dedicated circuit, you could mount the inverter close to the battery and run a smaller wire to the wires that now feed the AC to the TV. Just make sure you interrupt the circuit so you don't have the possibility of two 110v circuits trying to power the same item.
