Theldb
I also checked for you, at our FH library. by
"Index 1784-1914 of BMD in Newspapers, Bermuda" by C F E Hollis HALLETT
There were 3 male births, both unnamed, for Wainwright c1876. But I don't think they were yours - in fact one wasn't, and the other said father's name was John. Also, neither said Hamilton, but I don't know the geography in Bermuda so they maybe are near Hamilton? They were
Publ. 2 Feb 1875
John N. Wainwright, birth of a son on Jan 31 at Waterloo Cottage, Pembroke
Publ. 1 Feb 1876
to Willia R S Wainwright, a son on 16 Jan, at Halifax. However...
Publ.29 Feb 1876 - death of son of Willis R. S Wright's son aged 19 days, on 4 Feb, at Halifax
Also Publ. (forgot to write date) a birth to Willis Riddick Stowe Wainwright, a son on 27 May 1874 at 7 Brenton Place, Spring Gardens, Halifax.
Also searched Hallett's CD of "19th Century records of Bermuda," 2nd ed, both for Franklin and Wainwright - this is probably where fairyoevil looked - and found lots of Wainwright people and some wainwright occupations, but no Robert Franklin.
There is also a book "Early Bermuda wills 1627-1835" compiled by C F E Hollis Hallett, with 3 pages of Wainwright references - wills, executors, witnesses, etc. but I didn't check any names as I was not sure when there, whether the family had been in Bermuda long enough to have wills there.
Dawn M