I still can't get the DS4 to work in the game at all. I have it paired to Windows 10 through bluetooth, and Steam recognizes the controller. I've tried checking and unchecking the PS4 controller support and restarting the controller in between, and launched the game from Big Picture mode as well.
It's probably Origin getting in the way. It works to navigate around Big Picture but button presses aren't doing anything in game.
Edit: I got the controller to work by going under Manage Game in big picture mode and changing the controller options to Force On. It's still showing Xbox controller buttons though in game, but it's a start!