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Thank you for looking on my behalf. That link you showed for Mauritius BMD is amazing, with so many names I felt sure something would pop up there.  Trying to think what other scenarios would be appropriate.  I think it most unlikely that Rose Snr left the island  taking baby Anne with her and leaving  two older daughters - but who knows.  I do know that when William re-marries he describes himself as a widower.  Though I suppose he could say that even if his first 'wife' was alive, if he felt sure she would not return to England.
In 1861 Rose Jnr was living with her aunt and uncle.  She always used their name. They regarded her as their adopted daughter.  My family have a couple of samplers she made at this time on which she has embroidered Rose Desmond.  I guess when she was planning to marry she told her father. On the occasion of her marriage she would have been asked the name and occupation of her father so I suppose that is why she is registered as Rose Desmond Bennyworth.  I doubt if William was at the wedding. Rose's aunt, Mary Ann Desmond was a witness.  Only once more does Rose use her Bennyworth name and that is when her first child is born.  If it were not for these certificates I would have had no idea of the identity of her birth father.
It has been gratifying to be able to trace the Bennyworths back in time in Lincolnshire but I think Rose's birth mother is destined to remain a mystery for ever.

Hi Dawnsh,
Thank you for your suggestion. I don't seem to be able to find a means for searching overseas regimental deaths. Is it through the GRO website?
I know that William Bennyworth, serving with the 5th Foot regiment in Mauritius had two daughters while he served there.  Rose born March 1848 and Anne born 1851.  Prior to his posting to Mauritius he was stationed in Ulster and had a child Mary Agnes born in July 1845. I have been trying to find out the name of the girls' mother/s.
I have the birth certificate for Rose and on it her mother's name is given as Rosa - but no maiden name is there.  I ordered the certificate for Mary Agnes (born in Cavan Ireland) and disappointingly her mother's name is not given at all.  On the basis of this I am reluctant to order Anne's certificate if that carries only similar information, or lack of.
By the time the 5th Foot Regiment left Mauritius in 1857 William had retired on a pension back to his home county of Lincolnshire and here he remarried in 1856.  His marriage certificate describes him as widowed, so I assume he was previously married to the girls' mother.  This assumption means that Rosa Bennyworth probably died in Mauritius and maybe baby daughter Anne did too as neither can be traced back in Britain.  The older daughter Mary A went into service on return to England and married and the middle daughter Rose went to live with a William and Mary Ann Desmond whom she called aunt and uncle.   William D had also served in Mauritius and he and his wife had no children and appear to have adopted Rose. I haven't been able to prove that they were blood relatives 
Sorry, this sounds very convoluted.  I keep returning to it every couple of years but doubt if I shall ever know who Rosa was.
Any bright ideas gratefully accepted 

Has anyone had any experience of obtaining death and burial records from Mauritius.  Do they still exist? If they do, how best can they be searched without going to Mauritius?

Thank you. I can see I will have to make a visit to the Maritime Museum.  Is it best to notify them before hand if one wishes to see specific documents? 

Thanks Ady and Seaweed. The Joe Olsen born 1894 is born too late, but Joseph A Olsen (1878) is the man.  Bengal certificate241 issued Bengal 1902, have come across this reference before.  Been thinking about planning a trip to the National Archive so it is useful to know I might obtain more infomation there.  Can the Lloyds Captain's register be viewed at the London Metropolitan Archives? Very useful information
Thank you