Would I lenite or eclipse Banrion if I put "With the Queen" into Irish?
Technically, the definite article would cause lenition and the pronoun would cause eclipsis, but I assume that one would take priority here.

There's no pronoun. There's a preposition: le, which has a special form before the article: leis.
The preposition causes no eclipsis. Most prepositions (ag, ar, le, ó, as, faoi, roimh, etc.*) cause the usage of a "dative" form of the article which causes lenition or eclipsis of the noun depending on dialect: lenition (due to original dative form of article), eclipsis (due to original accusative form of the article).

leis an bhanríon (in Ulster and Achill)
leis an mbanríon (in Munster and most parts of Connacht)

All this doesn't depend on gender:
an bhanríon -> leis an bhanríon/mbanríon
an bád -> leis an bhád/mbád



*) except for three prepositions: de, do, i in Standard Irish. There's always lenition: den/don/sa bhanríon
With a notable exception of the exception: In Connacht (except Achill) there's sa mbanríon.

I don't disagree with anything Labhrás has said, but I think it's necessary to clarify a few things, as I don't think the kernel of your query was addressed.

In instances of preposition + definite article + noun, the initial mutation required by the preposition + definite article takes precedence over the initial mutation required by the definite article alone.
In this case, le + an + banríon = leis an mbanríon (except in Ulster and Achill Irish, as Labhrás has mentioned).

Note that the initial mutation required by the combination of preposition + definite article is not necessarily the same as the initial mutation the preposition would cause without the article. For example, compare:
le + Aoife = le hAoife but le + an + banríon = leis an mbanríon
i + gairdín = i ngairdín but i + an + gairdín = sa ghairdín




This is, of course, identical to the prepositional pronoun, 3rd singular, masculine form of the preposition, le, so the conflation of the two is reasonable. However, given its usage here, it can only be, as Labhrás has said, the special prepositional form used before the article.




I think Vitaee was referring to instances of definite article + noun, in which case the noun's gender certainly would require lenition for feminine nouns.

Thanks for the replies. I actually meant preposition and not pronoun in my original post, but my hands decided to ignore what my brain wanted to type.
I had the feeling the answer would be "it depends on which dialect".

But to summarize:

leis an mbanríon in Connacht and Munster
leis an bhanríon in Ulster and Achill

Correct??

I might add that throughout the Caighdeán Oifigiúil, the urú form is used.

But the Caighdeán accepts the use of the séimhiú, even though most standard texts use the urú.

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