I just started playing the HD collection and had trouble with getting the game to recognize my DualShock 4 controller. It didn't respond to any button presses with Steam or DS4Windows (an app for general DualShock 4 support and settings) running, and I was ready to just forget it and use my keyboard when had a thought to try using DS4Windows to emulate an Xbox 360 controller. The HD collection now recognizes my controller and it works just fine, though the buttons are all wrong of course.

So, in short, if you are having the same problem with your DS4 controller not working with the game, perhaps you could try emulating it as an Xbox controller with DS4Windows.

For other controllers, try X360CE or XOutput as well. Installing Vigem Bus Driver and HidHide is also recommended.

Edit: I also added the game (dmc1,2,3 exe) as non-Steam game and enabled Steam Input on the Properties which made the controller work but I think GOG overlay and achievements won't work with this method.