Well, Dylan, let's take the Protestant claim first. I'm always amused when I hear them speak of "the pure Gospel of Christ". Wait, is that the Lutheran, Calvinist, Episcopalian, Methodist, Baptist, Anglican, Assemblies of God or Seventh day Adventist's "Gospel"?
Your site says "had not the missionaries of Patrick's Celtic Church brought the uncorrupted Gospel not only to the rest of the British Isles but to Europe", but this is another fond Protestant fiction. Not only was St.Patrick unequivocally and demonstrably Catholic, the Apostle of the Emerald Isle was originally sent there by Rome, commissioned specifically for the charge by Pope St.Celestine I. Likewise, Rome was instrumental in sending St.Augustine (of Canterbury, not Hippo) to England, St.Boniface to Germany etc, all of whom brought the light of Christian Faith to these lands. In fact, almost all the West was evangelized by missionaries directly or indirectly from Rome and used the Latin rite, as missionaries sent to parts of what is now Russia from Constantinople used the Byzantine or Greek rite.
The 1172 Synod of Cashel and its acts is reported by Wiki here
. As you can see, it contains no formal abjurations of heresy that would have been required by Rome had the Culdee monks believed the "Gospel" of the Protestants. Rather, most were disciplinary, some concerned abuses, some related to morals etc. So these were Christians who more or less were still in communion with Rome but had fallen into decadence in some areas, did not pay tithes etc.
And just consider the example of St.Malachy a Saintly Archbishop who lived in Ireland who died 24 years before this Council, who knew St.Bernard, visited Rome, reformed discipline and morals etc and in every way showed that the indigenous Irish Church in general was conscious of and depended on communion with Rome and the universal Catholic Church unlike what the Protestant writer maintains. You can also read here
that some local lords themselves appealed for or approved of Henry II's actions. This was therefore not "an abandonment of the kingship" but something that was in their own mutual interests.
Compare all this with Henry VIII. He broke from Rome and founded a new nationalist sect, divorced and remarried several mistresses, and compounding all his prior injustices, England not only launched a bloody invasion of Ireland, but even suppressed native dissidents roughly. What St.Patrick and other Catholic missionaries had gained for the Church from the world wrapped in pagan darkness by right, prayer and preaching, the Protestants usurped from the Church by wrong, power and oppression. So even quite apart from the fact that heresy has no claims over truth, there is no possible grounds for any comparison such as that your friend makes, one only for stark contrast, in Ireland, England or elsewhere.
Besides all this, coming to the erroneous claim that religion cannot be a precondition for princes, that is a species of secularism. And it is based on the false premise that truth and error are to be esteemed the same, and is therefore itself an error. One cannot be "neutral" between truth and error anymore than one can be such between light and darkness, for that is blindness itself.
A country's Christian values cannot be peripheral to its civic life, much rather it must be foundational, the Bible held more dear than any human constitution however noble, the Fathers of the Church esteemed more than all founding fathers however wise, and the judgment of the Magisterium more binding than that of any court however learned, and its kings finally truly confessing Him who is by right their King, the Prince of Peace.
Religious liberty for adherents of error can be granted to the extent of practical toleration.
Finally, coming back to St.Malachy, he foretold that all this tumult between Ireland and England would end gloriously for the Faith.
"This prophecy, which is distinct from the prophecies attributed to St. Malachy concerning the popes, is to the effect that his beloved native isle would undergo at the hands of England oppression, persecution, and calamities of every kind, during a week of centuries; but that she would preserve her fidelity to God and to His Church amidst all her trials.
At the end of seven centuries she would be delivered from her oppressors (or oppressions), who in their turn would be subjected to dreadful chastisements, and Catholic Ireland would be instrumental in bringing back the British nation to that Divine Faith which Protestant England had, during three hundred years, so rudely endeavoured to wrest from her."