Judging from the wire diagram, without lights (including brake lights), you don't need the switch at all.
The key switch is switching the red wire to black when 'on'. The black wire is only feeding the horn and brake lights.
The 'off' position is connecting the ignition (black/white) to ground, cutting off the power to the ignition coils.
The kill switch ("Engine off switch") does the same, so if you keep it, you don't need the key switch. But then you'll always have to use the kill switch to stop the engine.
If you want to get rid of it as well, you can roll your own by connecting black/white and green to your toggle switch. The engine will stop if you switch it on