I'm not familiar with the " Keyless Go " system on this car. I'd expect if the doors automatically unlock on approach / handle pull, that should be enough authorization to crank / run the engine when the start button is pressed.

You still need to check voltage starting at the battery and various branches. It is possible that a branch has a bad connection causing the problem. Be sure to check for voltage while trying to crank as voltage can be good without a load then drop dramatically with a load.

Having a wiring diagram is a vital next step.