Are we taught t prothesis explicitly at some point? From what I've read it's vowel-initial masculine nouns in the nominative. I'm never sure, when I see nominative for Irish, if that's supposed to include direct objects or not (which might be accusitive in many languages). It's been explained how the forms merged, but for some reason won't stay in this dumb head. A purely synchronic explanation would probably help. A lesson or two on t- would be great, it's a minor mystery to me !
I believe the DL pronounciation sounds like the Connacht one here: https://www.teanglann.ie/en/fuaim/an_tAontas_Eorpach
The Irish for "Union" with a capital "U" is "Aontas" with a capital "A". When a particular grammatical circumstance calls for an initial mutation of any kind, the capitalization does not change, even when the mutation precedes the root word, but any hyphen that might occur is dropped.
"Uachtarán na hÉireann", "i gCeanada", "Ár nAthair".
As an adj., it's used attributively (not predicatively) here. So it must be declinable.
Teanglann.ie states that it belongs to the 1st declension of adjectives.
Nualeargais.ie adds that
the 1stD. is composed of adjectives ending in consonants (with the exception of those ending in -úil or -ir);
generally, these adjectives are declined like the nouns of the 1stD. (m.) or the 2ndD. (f).
So Eorpach, as a m. adj. , should follow the rules of the 1st.D. of nouns.