"the Irish government needs to be involved [...] if the data doesn’t clearly belong to a US citizen
No. Actually the CLOUD Act simply tells the involved company may file motion to a US court to quash the request if it believes the data belongs to a non US citizen.
Then a US court decides if the request should be quashed, taking into account the interests of the United States
A foreign court is never involved. That was exactly the aim of the act, otherwise it would have been exactly like before.
Moreover read what "qualifying government" means in the act - because if it is not a "qualifying government", there are even less protections.
Also, you can only "qualify" on US terms - US never sough an international agreement or treaty, just made its own law and wants to force others to abide. Hope they will be shown the middle finger.