'Sí an tsráid is cúinge ar an mbaile í
or
'Sí an tsráid í is cúinge ar an mbaile
are equally correct here. Note ar an mbaile not sa bhaile.

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The dialect I use is Cork Irish.
Ar sgáth a chéile a mhairid na daoine, lag agus láidir, uasal is íseal

Ah, I see! Go raibh míle maith agat, a Lon Dubh!

According to C. Ó Cadhlaigh in "Ceart na Gaedhilge" (a small and old but very recommendable book about Irish syntax), the article can be omitted in the former but not in the latter:

Is í sráid is cúinge ar an mbaile í.
Is í an tsráid í is cúinge ar an mbaile.

Cormac Ó Cadhla was not a native speaker of Irish, but an L2 speaker, albeit with an expert command of the language and a chair at university.
Peadar Ua Laoghaire wrote "Isé rud é is éagsamhlaíghe agus is tuathalaíghe agus is mó neamhthuisgint d'ár airígheas riamh".
The reason for the dropping of the article is that the predicate is contextually defined (the is éagsamhlaí etc clause defines it). I explain this in my article on the definite article published in Éigse.
Now Ó Cadhla's rule might relate to some other people's Irish other than Ua Laoghaire's, as there is and was always variation among native speakers.

Part of me doubts this. Him and Ó Nuallain both heavily based all their books on trying to make sense of everything Ua Laoghaire (or Keating too, in the sense of Ó Nuallain) wrote. It's actually part of why I really dislike them, even though they'd probably be helpful to my understanding of Irish; they're heavily biased towards the idiosyncrasies of one or two people (none of who use the dialect I'm focusing on), but they play it off as if this is how True Irish should be.

Well Ua Laoghaire is the most prolific writer in any modern Irish dialect, and the Cork dialect was a continuation of the 18th century literary tradition in a way that the Ulster and Connacht dialects are not. Of course from a descriptivist view, all dialects are equal and any can form the standard, but the claims made for Muskerry are correctly made. The CO approach focuses on the Irish of precisely ZERO native speakers.

In any case, Cormac Ó Cadhla's statement on the copula here is at variance with Ua Laoghaire's usage.

Was he the most prolific? Máire wrote quite a lot, and I wouldn't be surprised if he wrote more than PUL overall.

But, it is still quite annoying when they say "This is the only correct way" and give no attention to the other dialects, or even consider them. Specifically when those dialects are (and even were at that time!) more widely spoken. Simply because PUL had written a lot when they started writing. As I said, it sometimes seems like they're just trying to shoehorn everything that came out of PUL's pen into a formula, some of which seem, to me, to require some really awkward explanations to make it fit their system (speaking mostly of Ó Nualain here).

But, that's just my opinion of them; it's also why I don't rely on them or recommend them to anyone not focusing on Munster Irish (or even more specifically on Muskerry Irish)

In pseudo-clefts (a somewhat different construction), "is é rud é" is very common as an introduction and so probably affects other uses. It could perhaps be a kind of contamination in a normal identification sentence.
Do you know of examples with other nouns than "rud" or "duine"?

Yes: Isé galar é is measa a bhí ar aon Chríosdaidhe riamh.

In my published article on the article, I wrote:

So, as you note, Labhrás, although this is used with all nouns in Ua Laoghaire's Irish, it is a recognised thing that in Northern and Western dialects, this usage is only found with rud.

Do you have an example? Most of what Ó Nualláin's grammar says seems fairly natural to me, i.e. the explanations don't seem like that much of a stretch.

Regarding dialect, I thought they were fairly middle of the road. For example Ó Nualláin's section about the copula doesn't point out Munster specific features. Are there Ulster or Connacht forms that they label as incorrect, claiming only the Munster form is right? Even in the cases where they explicitly invoke a rare type of sentence from Ua Laoghaire, it's often as a preamble to explain a type of sentence common in 19th century Irish in general. I'm willing to be corrected here, it's about a year since I read through them, so I probably don't remember everything. Most of the cases where they say something is wrong that I remember were about learner errors.

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The dialect I use is Cork Irish.
Ar sgáth a chéile a mhairid na daoine, lag agus láidir, uasal is íseal