To answer this question, you would have to ask why a Welshman or a Scotsman would be acceptable where an Irishman would not. The answer is religion.

"The middle decades of the nineteenth century represent the high water mark of anti-Irish prejudice. The reasons for such intense discrimination against the Irish were rooted in a past that predated the English settlement of North America. One major reason was the English cultural tradition that for centuries regarded the Irish as an inferior people. Evidence for this tradition dates back to the late middle ages.

Historians have written a great deal about this theme in Irish history.. It was part of the transatlantic cultural exchange that the English exported to the American colonies. The best evidence for this was the penal legislation of the eighteenth century. A key aspect of this legislation was religion. This was a second reason for anti Irish prejudice. Linda Colley has argued very convincingly about the “absolute centrality of Protestantism to the British experience in the eighteenth century and long after... .” She wrote that “Britons defined themselves in terms of their common Protestantism as contrasted with the Catholicism of continental Europe(and very often in contrast too with Catholic Ireland).” In British America the centrality of Protestantism to the people’s identity was equally evident. This was visible in the penal legislation of the early eighteenth century that endured until the revolutionary war of the 1770s. Each of the thirteen colonies in British America had some type of legislation that discriminated against Catholics, the vast majority of whom were Irish.

To be Catholic in the United States in the 1840s and 50s was to be portrayed as a menace to national security. To be Catholic was also to be Irish. At this time they were synonymous. This religious bias against the Irish reinforced the cultural prejudice that the heirs of British America carried with them well into the nineteenth century." (from an essay on "Anti-Irish racism" by Jay P. Dolan).

So, to conceal your origins by claiming to be Scottish or Welsh would essentially be claiming to be a Protestant. The Scots were Presbyterians and the Welsh were usually nonconformist or Methodists. The "Scots-Irish" or Ulster Scots, who had settled in the American South, were acceptable to the Anglo-Saxon American ruling classes because they were strong Protestants, going on to provide several American presidents.

Someone like, say, Oscar Wilde, an upper-class Anglo-Irish Protestant, visited America in the 1880's and was very well received. The prejudice was not just against the Irish, but specifically against the poor and Catholic Irish.