Unfortunately, I don't know anything about Joseph Bentham. I only know of Jonah Bentham and I wonder whether they are one and the same.
My Jonah married Margaret Elizabeth (no surname) and had a daughter, Jane Eliza, chr. 12 Sep 1797, St Philip's, Barbados. She was known as Queen Bess, said to have been a luxurious wealthy lady with many slaves. She married in 1819, Robert Burgess Garratt, who came from Bristol as a schoolmaster and later a planter. They had 6 children probably all born in Barbados. The youngest, Robert Bentham Garratt was manager of Clonbrook estate, East Coast, Demerara in 1892.
Trotman was a well-known Barbados family and one was in BG in the 1870s married and had a large family, but I know nothing of any extra-marital connections with Bentham!