I think the above would be:

Beannachtaí na Banríona Mhór

Or would there be a better way of saying this? If it matters, I'm referring to the Morrigan.

Beannachtaí na Banríona Móire.

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Is fearr Gaeilg na Gaeltaċta ná Gaeilg ar biṫ eile
Agus is í Gaeilg Ġaoṫ Doḃair is binne

Yes

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I'm not a native (or entirely fluent) speaker, so be sure to wait for confirmations/corrections, especially for tattoos.

Whoops! I assume that "Móire" is the genitive singular female which it of course should be in.
Táim amaideach! I forgot that adjectives must also agree in case.

But, is there a better word to use here than "Móire"?

If you mean it in the sense of "high queen" or "arch-queen", you could presumably use the prefix "ard-" instead, giving you:

Beannachtaí na hArdbhanríona

The prefix ard- causes the lenition in banríon [i.e., "b" becomes "bh", with a "v" sound], and the "h" before the word is there (in writing and in pronunciation) to smooth the transition from the word "na".

I said that this will presumably work, because - although the word ardrí ("high king") is common enough - I've never actually seen the word ardbhanríon.

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I'm not a native (or entirely fluent) speaker, so be sure to wait for confirmations/corrections, especially for tattoos.