The problem with both Normans and Scandinavians was they really, really quickly blended in the Irish situation, culturally as structurally.
and it's not certain that William the Conqueror, the most well known of the Norman leaders, had any Viking heritage himself.
You mean, apart being the direct descendent of Rollo? That said, I agree with you, while the Celtic element in High Middle Ages France is the kind of element you can see with a microscope only. Normans themselves, at this point, used "Franci" as an ethnonym.
Now, how to get Barbarians in Ireland? Well, if how they get in Britain in first place is any indication, you'd need to have some need for manpower : both for economical and military purposes.
Problem, Gaels were more about raiding themselves, and were "protected" by Britain that served as an insular shield.
Eventually, you'd need Hibernia being conquered or hugely clientelized by Romans ITTL (maybe after Caledonia), and seeing the Saxon Shore being extended to southern Ireland, maybe briefly before seeing Ireland being evacuated by Romans.
I don't expect Anglo-Saxons being much interested on Ireland, to be perfectly honest : much of what made its wealth was due, among other reasons, being partially isolated from Romania and being able to raid (that said, Gael-Saxon raiding parties could still happen ITTL).
Eventually, you may have a situation where Ireland appears as the natural continuity of Britain, especially if a Dal Riata Gael-Saxon kingdom is established on Western Britain.
IOTL, Northumbrians made a raid against O'Neill in 684, mostly due to succession issues. So it's not technically unthinkable to see Anglo-Saxons acting against Gael-Saxons as well, at least against this Dal Riata equivalent, and possibly against Eastern Ireland, especially if they consider it as running on from western Britain.
Eventually Gael-Saxons would probably being much Gaels in many matters, but with enough geopolitical closeness from Anglo-Saxons to be more "Saxonised" up being Germanised.
It's a TL that would ask for a lot of refinement to work, but may be made more or less plausible enough for being believable.