As far as I'm aware, the only records of death before 1837 were burials from the parish registers, so unless the vicar was one who kept really good records ie added more than he was required to, there probably isn't any hope of finding cause of death.
Is there a grave with a headstone, that might prove whether or not the child was hers? Strange there was no baptism, usually one was done quickly if a child was expected to die.