Enabling recompute tangents on a skeletal mesh does not recompute tangent normals for translated parts of the skelmesh.
If you add
r.SkinCache.RecomputeTangents=1 Do not access these cvars directly
to the console variables.ini Recompute tangents will correctly work.
There's no existing public thread on this issue, so head over to Questions & Answers just mention UE-38398 in the post.