How can a capacitive touch screen be triggered without human contact?
I haven't actually done this, but it seems the problem is the objects you are using are too small and don't have enough ambient capacitance.  A human touching something adds capacitive coupling to the environment.  Think of the size and surface area difference between a carrot and a carrot+human.

You should be able to use something conductive that is covered by a thin insulating layer, then connect the conducive part to a conductive plate under the iPhone or to ground.  In this case "conductive" only needs to be not a good insulator.  As you found, even something like a carrot is conductive enough.  Try connecting a ground clip to the other end of the carrot, or connect it to the chassis of your machine.