Might the English verse form have gotten its Irish name in America? Maybe, maybe not, but consider the entry on Limerick in the Historical Dictionary of American Slang (Jonathan Lighter, editor). “Come to Limerick”–only in American slang–used to mean, more or less, to settle, to come to terms. Members of the American Dialect Society discussion list added to the three examples given in the Dictionary. They range in date from 1859 to about the end of the century. The first uses mostly relate to the looming and then raging U.S. Civil War; they later referred to more diverse put-up-or-shut-up situations. (More details to come in the comments, if interested.) There were many “Limericks” published then in America. I suggest the reference was to the end of the earlier Irish Civil War that was partly concluded with the Treaty of Limerick.
The OED quotes J. H. Murray–not to be confused with J. A. H. Murray, the OED editor (proof available on request)–in 1898 writing in Notes & Queries that Limericks were offered at convivial parties with the “come [up] to Limerick” chorus sung as a challenge for a new verse: in effect, offer another new one or surrender.
Admittedly, the above does not prove an American origin of the name. But here’s another hint that the name did not refer to poets literally and literarily from Limerick, having left, as the Treaty allowed, in the Jacobite “Flight of the Wild Geese” to France. In 1881 the Church of England Bishop of Limerick, who was also a poet (and relative of author Robert Graves), received an honorary degree from Oxford. This is recounted by his son, also named Charles Graves, in “The Cult of the Limerick,” Cornhill Magazine, Feb 1918, 158-66 (here 158):
“…he [the Bishop, in June, 1881] was greeted in the Sheldonian by cries of “Won’t you come up, come up, Won’t you come up to Limerick town?”–which we believe to be the correct form of the refrain. But the reason for the connection of the City of the Violated Treaty with this particular form of pasquinade remains, as Stevenson said of the young penny-whistler, ‘occult from observation.'”
(I have added the italics.) There is, of course, good stuff in the comments.