It might be an idea to expand the search for the mother over a larger time frame. As well as a death, look for a marriage.
It may have been seen as an easier option or a way to protect children to tell them that their mother had died (with no more questions asked), rather than tell them that their mother abandoned them or went off with another man. I am not implying that this is what happened here, but as per Lizzie's example, family stories can sometimes be wrong.
If you can't find a second birth registered for the "twin" it may be that this was invented to explain the mother's death or disappearance?
Have you found proof of the father's death at sea?
Sorry to appear negative, but as you can't find proof of the birth and death, that made me a little suspicious.
Have you explored the possibility that the birth or death might be registered under a different surname? As you say the name is unusual, have you checked for spelling errors?