With UpdateLayer, are you using False for the symbology_only parameter? It defaults to True, so the label properties will only come across if you specify it as False.
For example: arcpy.mapping.UpdateLayer(df, updateLayer, sourceLayer, False)
Keep in mind this will bring all layers properties including symbology, font, font size, label expression, etc., so everything will need to be defined in the update .lyr file.
David