Easiest way is keep the inverter close to the batteries (otherwise you’ll need larger cables). Then install new receptacles wired using an extension cord with the female end cut off. Then plug the male end into the inverter. This will avoid any need for transfer a transfer switch or relay.
The TV is likely already a 12-volt unit. If it is, you can simply remove the transformer power supply and hardwire in the 12-volt lead. This will avoid the loses of stepping up then back down, and modified sine inverter can create noise in A/V equipment.