Valet Key Start?
 

Can a push button start BRZ be started with the valet key? If so, where is the valet key inserted to accomplish this feat?

The physical key is not a valet key. It's exactly the opposite. You're supposed to give the valet your clicker and take the physical key with you.

The reason is the trunk lockout. The valet button in the trunk locks it so that the valet can't get into it using the clicker or the button on the dash. You leave the electronic key with the valet and take the physical key with you, which you then use upon your return to unlock the trunk and turn off the valet button.

In reality the valet button is a useless novelty, as it is a very simple matter to pull down the rear seat from inside the car. But the fact still remains that the system is set up for you to leave the clicker with the valet while keeping your physical key on your key ring in your pocket.

And no, you can't start your car with the physical key.