Doing some investigation online it is definitely possible to use PS4 controllers (including DualShock 4) on PS3. Please read this link (one of many similar guides) for more details on how - relevant excerpts on the procedure described in the link follow:
"To use a PlayStation 4 controller with a PlayStation 3 console, the easiest approach is to simply connect the controller using a USB cable. Once the cable is connecting the two pieces of hardware, you'll be able to play PS3 games with the controller right away.
How to connect a PS4 controller to your PS3 for wireless use
- Connect the controller and console with the USB cable and power both on.
- Open the Settings menu, then go to "Accessory Settings."
- Click on "Manage Bluetooth Devices."
- Select "Register New Device" by pressing "X" on the controller.
- Click on "Start Scanning," and then disconnect the cable from the PS4 controller.
- Hold the "PS" and "Share" buttons on the controller, and when the lights on the controller start flashing, plug the cable back into it.
- Highlight "Wireless Controller" on the screen and click on it.
You should now be able to use your PS4 controller wirelessly with your PS3."
Failing that according to this post (found on this same site titled "Can you setup a PS3 without a controller?") it would appear your next best option is to perhaps borrow a controller that can connect to the PS3. Unfortunately, I cannot find information on how you could go about "hacking" the interface to get it operating another way.