In the 1655 expedition, England's target was actually Hispaniola. Only after the Santo Domingo campaign failed did the English, facing disgrace at home, decide to attack Jamaica. If they capture Hispaniola then the Jamaica campaign is butterflied away.
French pirates were already based in west Hispaniola at this time. Perhaps France and England make an accord so that the French leave Hispaniola in return for England's assistance in capturing Jamaica.