Why did you use a USB host shield if the Teensy 3.0 can do USB OTG? You could have saved money by just getting a USB OTG cable like this http://www.amazon.ca/Micro-USB-OTG-2-0-Adapter/dp/B005QX7KYU and save yourself some money and a lot of room by not having the host shield, plus it would look much better, one less hole to cut in your DS4
Teensy 3.0 hardware supports otg, but not software. It is not done yet and the last update i read was that not a single line of code is written for host mode support on teensy 3.0.
I'm doing something similar, I am using STM32 as a MCU, which has two USB interfaces that are both capable of OTG, nice eh? I already replicated XIM3's features with it but PS4 is a different story.
ST makes STM32 and they provided most of the USB host code, I added my own hub support to it. Teensy 3 uses Kinetis from Freescale and I think they also provide a USB library, perhaps as a part of their MQX library.
Anyways I just wanted you to be aware of some alternative solutions