This line is a fine example of situational irony and rhetorical positioning. Readers who take this line literally might expect Swift to actually offer a modest idea. However, the second sentence, which claims that no one will object, points to the irony of the situation and the proposal to be followed. Many people, including Swift, had tried to solve the problem of Anglo-Irish relations. None of the solutions were successful. There has also been considerable active debate about what to do. Swift made several suggestions. All were rejected. To say that no one would oppose it is a great reversal of the current situation, a kind of ironic exaggeration. That one long sentence is the whole second paragraph.
Swift creates common ground with her reader, claiming that everyone in Ireland agrees that so many poor beggars in the country are a serious problem. The reference to the islands of Barbados is a comment on the fact that many Irish who could not feed themselves immigrated to the New World. I think all parties agree that this astonishing number of children in the arms or on the back or on the heels of their mothers and often their fathers is in the current deplorable state of the kingdom, a very great additional complaint; and so anyone who could find a just, cheap and simple way to make these children healthy and useful members of the Commonwealth would win so much of the public that his statue would be erected for a guardian of the nation. Leave their dear homeland to fight for the Pretender in Spain, or sell yourself to the Barbados. Swift is referring here to England. .