How can I get my PS4 controller to work for Legacy of Kain: Soul Reaver on Steam?

Since your PS4 controller already works as a XBox 360 controller, there already may be a solution to your problem:

https://steamcommunity.com/sharedfiles/filedetails/?id=497805274

If this doesn't work, I would suggest Motionjoy gamepad tool or xpadder.

InputMapper isn't needed now that Steam has its own DS4 support built-in. Try uninstalling or disabling InputMapper and configure Steam to enable its DS4 support.

From what you described I think the gamepad is not calibrated correctly and has no deadzone. Steam should fix that for you, but it might also work to calibrate it through the USB Game Controller tool.