In XNA there are methods:
TextureCube.SetData(CubeMapFace cubeMapFace, int level, Nullable<Rectangle> rect, T[] data, int startIndex, int elementCount) (ref: https://msdn.microsoft.com/en-us/library/ff434428.aspx) and
TextureCube.GetData(CubeMapFace cubeMapFace, int level, Nullable<Rectangle> rect, T[] data, int startIndex, int elementCount) (ref: https://msdn.microsoft.com/en-us/library/bb197085.aspx).
There you can specify the mipmap level and copy the mipmap texture data to the cubemap, it’s very easy. 
BTW: the second method is missing from the original MonoGame source, but it’s pretty straight-forward to implement it.
EDIT: Updated the post, because the method TextureCube.GetData(…) is missing, not TextureCube.SetData(…), my mistake.