Based on what I see in this build, the Dark Rock 3 fits really well, making me think that the Pro may have a little room.
https://pcpartpicker.com/b/tX4CmG
This guy is also using the Pro 3 with a Define R4, which I THINK is about the same size as the P400S.
https://pcpartpicker.com/b/xWgLrH
You may have to remove the exhaust fan completely to fit it, but that's just a sacrifice you're going to have to take.
And besides, why do you have to have a Dark Rock Pro 3? There are plenty of other great coolers on the market. You could also get a new case. Your choice though.