I personally think Catherine (Parker) was Roberts sister. The childrens names - Isabella & Lydia - both fit my relatives, and my gradfather, one of Isabellas was called George Beausire. I agree it is odd her being born in Liverpool but its also odd they the family decamped to England as I understand it around 1810.
I thought the Beausire from Lowensteins Chasseurs who was wounded was hurt on St Vincent? I will trace the Google connection you gave me. However the fact that Joseph came from Metz was a lead from an entirely different source, a Frenchman who also has strong links to the West Indies. The Parkers incidentally are almost certainly a union between a British navel man and a French? woman, with the children born on Martinique.