Not really, unfortunately. The Irish were very much looked down on, both the native Irish lords and the descendants of English lords who become Gaelicised. Any link-up with the Irish would raise questions in court. Also, the Irish were fiercely Catholic
Not entirely true. There was intermarriage between the English nobility and Anglo-Irish nobles into the 16th century - I seem to remember that Wolsey (to his later regret) stiffed an early love affair of Anne Boleyn because the guy was slated to marry the Earl of Ormonde's daughter.
As for religion, how fiercely Catholic were the Irish in, say, 1560? My impression is that Catholicism only became an expression of Irish proto-nationalism during Elizabeth's reign. (Remember that it was in Catholic Mary's reign that quasi-genocidal 'plantation' of Ireland was first proposed.) If Elizabeth marries an Irish earl early in her reign, the whole dynamic of Anglo-Irish relations is changed - you'd have an Irish king, and the founding of an Irish dynasty.
The practical problem is that no Irish earl was remotely suitable.