I don’t know if this still helps you, but if not, maybe the next person who has this issue will find it.
I’ve been using houdini to build destructible meshes (which sort of technically work as skeletal meshes, I think), and was having issues with broken normals.
The solution turned out to be unlocking the RBD to FBX node. I changed the fbx export to 2016, but crucially turning on ‘compute smoothing groups’.