This is from the Gospels:

How can cad é sin dúinn-na sin? be parsed.

If cad é an rud é sin? as O'Nolan says is PsS, where cad is the predicate, é the subsubject and an rud é sin the subject, then why are these two sin's in cad é sin dúinn-na sin? É sin can't be a subsubject, as only é would be. "Sin dúinn-na sin" can't be a subject with two sin's.

Cad é sin dúinn-na 'what's it matter to us'. I'm guessing dúinn-na sin is a pronoun + demonstrative adjective, like iad san, but then I would expect dúinn-na san. Could it perhaps be a typo for dúinn-ne? This is still a bit of a stretch, though, because wouldn't the use of the first person automatically cancel out the use of the demonstrative sin, at least, that's how it would be in English.

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I'm an intermediate speaker of the Corca Dhuibhne dialect of Irish and also have knowledge on the old spelling
Soir gaċ síar, fé ḋeireaḋ thíar

It was dúinn-ne in the original. I edited it with a broad -na in line with the pronunciation. This suffix is always broad. But why the two sin's - that really my question?

The only thing I can think of is that it is a bit like tá a fhios agam é, where Peadar Ua Laoghaire stated that the final é was required - the é can replace a whole clause (tá a fhios agam gurb ea, etc), and is required, but is non-intuitive to an English-speaking learner, and the a before fhios is proleptic.

Cad é sin dúinn-na sin? We can regard the sin as standing in for a clause/phrase. For example in Matthew 8:29, we read Cad é sin dúinne thusa, a Íosa, a Mhic Dé?, where the Douay has "What have we to do with thee, Jesus Son of God?" So the sin fills the same slot as the thusa.

And in Niamh: cad é sin dó san cad a dhéanfaidh an Árd-rígheacht? "What does he care what happens to the High Kingship?" where the sin slot is filled by a whole clause.

There is similar Cad é sin dúinn-na thusa, a Íosa, a Mhic Dé? in Maitiú 8:29
What have we to do with thee?

So, in Cad é sin dúinn-na sin?, the last sin is: … with that?)

The first cad é sin is probably kind of an emphasized what ("what exactly")

From Ó Cuiv's Irish of West Muskerry: Cad é sin adubhairt sé? Cad é sin ar siobhal?
Here it still has a seperate meaning: What is that what he said? What is that what is going on?

Trying a more literal translation:
Cad é sin dúinn-na thusa? = What is that what you are to us?
Cad é sin dúinn-na sin? = What is that what that is to us?

Thank you, Labhrás.