The problem with using accents is, unless you work on them individually in relation to the song and the area it came from, to the same extent as you would work on the tune, ornamentation, interpretation, phrasing etc., they can, and quite often do sound phony. It can be done but it takes hard slog, and if you don't do it they can sound like stereotypes or even piss-takes of accents.
Virtually all English people I have heard who attempt 'American' accents end up producing a mid-Atlantic approximation - neither fish nor fowl - which sounds sometimes hilariously phony to American listeners (sort of like Dick Van Dyke's Cockney in Mary Poppins) and can even be taken as insulting - our pop music is based on this phony pasted-on accent.
Accents can be beautiful (maybe not Birmingham – joke-joke!) - they are a part of our heritage and we stand to lose them to 'Estuary' English' or to the emasculated travesty of English our language is becoming on the media, so they are worth preserving properly.
On the other hand, if a song will Anglicise, satisfactorily why struggle with something unnecessary?
If it won't and you still want to sing it - do the work or you'll almost certainly make a hames of it.
Anyway, people don't sing in 'American' or 'English', or even 'Lancashire' or 'East Anglian' accents - our speech is far more fine-tuned than that.
There isn't one single 'Irish' accent, as pointed out - as sean-nos singer, Nell Ní Chroinín (above)
"I wouldn't sing songs from, say, Connemara because you need that blás, that local fluency".
If it's a problem for her – it's ten times more of a problem for us foreigners or outsiders.

"I've found that you pick up the accent and speech patterns of where you are living...."
I was born in Liverpool, I moved to Manchester when I was 25, lived there for four years, then moved to London and survived there – (nobody 'lives' in London!) for thirty years.
I have now lived in the West of Ireland for 15 years and will almost certainly pop my clogs here.
Now, in all the places I have lived, people ask me where I'm from – including in my birthplace, though most people invariably guess I've lived in Liverpool at some time or other.
Jim Carroll