The basic fallacy here appears to be the assumption that just because Ireland is one of the British Isles, its inhabitants, the Irish, must necessarily be "British". Now, I do not suppose there is any controversy surrounding the fact that Ireland and Great Britain are part of the same archipelago geographically? In this case, in order to claim there is a controversy, what alternative suggestions to "British Isles" are there as to the name of said archipelago? It sort of stands to reason that a group of islands may be called after the largest island it contains.  
 There are some suggestions, such as "IONA", but clearly none of them is an arguable competitor to "British Isles" within WP:NAME. Accepting that, I wonder what the    template is doing here. There may be a real world dispute, but the   template isn't intended to draw attention to real-world disputes, it is used for unresolved disputes on-wiki. Once a real world dispute is detailed fairly and exhaustively, there is no reason to keep arguing about it on-wiki.