Re: Is there ANY way to play this game with a PS4 controller?

You have to use an XInput wrapper to make the game recognise it as an Xbox controller. If you're on Steam, you can enable the DualShock in the controller settings and Steam will use its own driver to read it as an Xbox controller. If you don't use Steam, you can use third party software like DS4Windows.

You're going to get Xbox icons, though. Some games give you an option to pick your preferred controller icons, but I don't think Dragon Age Inquisition has that option.