Switch lamp with two serial switches
Indeed, you need to at the very least replace the circuit breaker to some sort of signal instead, as otherwise you wont be able to power the lamp. If you were to send the signal as a brief lack of power, you need some to make some bi-stable circuit in the lamp to do the switching.

I would recommend using a micro-controller and bi-stable relay. I would power the µC from a diode bridge and a capacitor, and use the EEPROM to remember the state of the device, and use the brownout/power reset to trigger a switch, i.e. to actuate the relay.

Now any simple push-button in a series chain (i.e. both your wall button and cable button) will trigger a toggle.

I hope you have space for a relay in your lamp-side circuit, as switching mains AC in any other way will be much more complicated...