That was correct! After adding “CellData to PointData” filter in ParaView I could export the scene correctly.
I will update this post and add the code once I figure out how to do it using vtkCellDataToPointData class. 
import vtk
cell_to_point = vtk.vtkCellDataToPointData()
cell_to_point.SetInputData(polydata)
data = cell_to_point.GetOutputPort()
# you can use this data now to set up the mapper
mapper = vtk.vtkPolyDataMapper()
mapper.SetInputConnection(data )
mapper.SetColorModeToMapScalars()
mapper.SetScalarModeToUsePointData()
mapper.SetScalarVisibility(True)
# ... more code to set-up the mapper as you need
# ...
# update mapper - I'm not sure if this is required but I'll do it to be safe!
mapper.Update()