Maya to UE4 - How to properly export smoothed mesh for fbx animation?

I’m working on an animation. I wanted to put my character mesh into Unreal Engine 4, so I can make a scene around it. The mesh looks like this in maya, and on the picture on the right, there is a static version of it subdivided and painted in 3d Coat - my endgoal:

However in Maya I have a problem with smoothing my mesh properly.
I’ve tried Modify -> Convert -> Smooth Mesh Preview to Polygons, but after exporting the fbx to game engine, it didn’t seem to work. The mesh was looking smooth in maya, but not in UE4.

So after some research I’ve tried to use Mesh Display -> Soften Edges. The result was mixed:

The mesh in UE4 looks smooth now, but for some reason beyond my comprehension the UV’s are messed up. While as you could see on a picture above, the exact same mesh without using Soften Edges was fine. At first, I thought there may be a problem with skeletal mesh having a higher resolution, but it doesn’t seem like that, because the static mesh is bugged in the same manner.

I’m lost, it seems like I’ve managed to solve one problem, just to cause the new one.

Please, send help!