the cb 360 right hand control have a return cable to the carbs,so 2 cables not 1 like the 350/450, you can also open it up and replace the button if you don't want to replace the whole controls,
also making a separate starter button is also pretty easy, just trace it back to the harness and get a button from canadian tire plus 2 wires,will do it, i've done a few that way,