There have been at least two attempts to rename a portion of the Protestant Episcopal Church as the Reformed Catholic Church. The first came in 1861 when the newly formed Protestant Episcopal Church of the Confederate States met to discuss its constitution. Bishop Green of Mississippi suggested the name, only to have to withdraw it when the strength of Virginia's opposition became known. The second came at the General Convention of 1889, and failed by only three votes!
Why the fascination with the name "Reformed Catholic?"
Anglicans - even the ones who if pushed would call themselves "protestant" - feel a certain discomfort with the term. The source of that discomfort is the fact that the word "Protestant" means not only someone who protests against the errors and innovations of Roman Catholicism, but it also covers a multitude of denominations that have chucked out any concept of being "Catholic" in favour of Radical Biblicism. However, inspite of being used with a capital-R to cover Calvinism, the term "reformed" better encapsulates what Anglican feel happened in their reformation. The Anglican Reformers took the existing Church and reformed it. As one Anglican apologist retorted to a Roman Catholic who asked "Where was your Church before the Reformation?" - "Where was your face before you washed it this morning?"
Anglicans understand their church to be Catholic because of its continuity. After all, almost all Anglican bishops have both William Wareham, the last but one RC, and Thomas Cranmer, the first Protestant Archbishop of Canterbury in their Apostolic succession. Anglicans also have the Bible, the Creeds, a reformed Liturgy, the Sacraments, as well as that threefold Apostolic ministry. Anglicans have also stated in the Chicago-Lambeth Quadrilateral that they believe the Scriptures, Creeds, Sacraments and Episcopal Ministry to be the marks of the Church.
So "reformed Catholic" reflects a great deal of the Anglican self-understanding as being in origin the Catholic Church of England reformed according to Scripture. Also, it isn't really a party label, the High Church Protestants of the eighteenth century used, as have some modern Anglo-Catholics, and in recent times, the Evangelical Scholar, Dr Peter Toon. The fact that it has been accepted as a describer for Anglicanism by men of various parties shows what a widespread appeal it has as a definer for Anglicanism.