Is the cemetery you checked, the nearest one to where the family were living when the baby died? Have you checked the cemetery records or just headstones?
My OH's great grandparents had two children who died in infancy, but they do not appear on the headstones in the cemetery where the rest of the family are buried. However when we contacted the cemetery authorities, they had records of the burials of these two babies in the group of plots where we were looking, but the graves are not marked.