It'll break down even faster if you add a bunch of N to it. :-D
It sounds like the cats are adding enough already.
It wont do much for the cat shit if it is too high in carbon, too high in nitrogen, or broken down too much. Plain 300-500-1 carbon wood chips aren't going to support the biology. A fine compost for mulch is not going to have the carbon or fungus to combat the poopies. Something composted with high N will break down fast but will also be missing the ever important fungal component. Fungus is both immune system for the ground, as well as the most efficient digesting body.