What you will have to do is wire the controller to the PC and override the controller settings for this game to disable Steam Input so that the DS4 controller's inputs do not get transformed to XInput.
Here are the steps on how to override Steam’s controller settings for this game:

1. Search your Steam library for the game. You can search for:
   "Black Ops III"
   
   
2. Right-click the game from your Steam library, then select Properties...
   
   
   
   
3. On the new window that opened, select the Controller tab.
   
   
   
   
   
4. Set the option for Override for Call of Duty: Black Ops III to "Disable Steam Input"
   
   

Once done, make sure the controller is wired to the PC then launch the game, and you should now have PlayStation controller button icons!

In the case that the previous steps do not make the game use PS4 controller icons, use DS4Windows or another controller emulator to emulate DualShock 4 inputs. This can be achieved on DS4Windows by doing the following:

1. Install and set up DS4Windows if you have not already, and make sure the controller is recognised on it: https://ds4-windows.com/get-started/#installation-setup
   If you are having issues starting the program or issues getting it to recognise your controller, check the following: https://ds4-windows.com/troubleshoot/
   
   
2. Once the controller is recognised by the program, select a profile for the controller which you would like to apply the changes to, then click Edit.
   
   
   
   
   
3. On the screen that opened, select the tab labelled Other.
   
   
   
   
   
4. Under Virtual Controller Settings, set Emulated Controller to "DualShock 4".
   
   
   
   
5. Once done, click Save to save the changes to the profile.

To get the game to recognise the emulated controller while playing, make sure to keep DS4Windows running during gameplay. Ensure to also follow the steps in the Instructions section of this guide if you have not already for the icons in-game to change properly.

Main menu


Gamepad button layout controls


Server browser


Multiplayer specialist ability/weapon


Zombies gobblegum

I would like to thank Distinger for informing me of how they got the PlayStation buttons to appear in the game, thus making this guide possible!