Don't mess with the breaker panel. If you want to put in a permanently installed inverter then put in a receptacle for it. Use a different colored receptacle and face plate so you don't get them mixed up. A 300 watt inverter will require about 30 amps at 12V so you would need to run a minimum of 10 awg wire to it. And be sure that there is a 30A fuse on your new 12V line to protect that wire.
But why permanently install the inverter? You can install a 12V power port, aka cigarette lighter port, and plug the little inverter into that as needed. If you get a 12V port with a couple of usb ports next to it you can charge your portable electronics that way too.
Also consider swapping the TV out for a dual voltage one. That's what I did, they're not expensive. Bolts right up to the same mount and will come with a 12V plug and a 120V one, so you can leave it plugged into both supplies and it will automatically switch to 120V if that's available. Much more efficient than running from an inverter.