So, I am a first language speaker of Irish, and I am studying the language and its literature and history as part of my course, and this is the best I can do with undoing the Anglicisation of "Ilvermorny", which, as @Bellatrix has mentioned already, is not a name that already exists in Ireland.
Ilvermorny < Ilver\morny < Oirbheart Muirneach
(trust me, it actually does sound like 'Ilvermorny' when said out loud).
It's kind of cute, I guess, because Oirbheart means, among other things, 'casting (of a spell)', and Muirneach, as an adjective, means ‘passionate, heartfelt or tender’. So, Ilvermorny/ Oirbheart Muirneach means (at a kind of stretch) "tenderly cast spell", which in my opinion fits into the canon of William and Ríonach Sayre deciding to help the Muggles in their area with magic.
Okay, so here goes my headcanon:
The Irish branch of the Gaunt family (ending at this point with Gormlaith and Ríonach) came to Ireland either because of feuds with the English Gaunts/other pure-blood families/etc. They had been in Ireland for a couple of generations at least, considering the Gormlaith and Ríonach's names (fun fact; Ríonach means 'a royal', and Gormlaith means 'Blue Princess' - very fancy names for SLytherin's descendants), and in my opinion, these two Gaunt sisters are most definitely bilingual - English and Irish.
HOWEVER, William Sayre is not an Irish Gaelic name. Rather, it's an Irish name that has been Anglicised. In terms of Irish history, that means that William Sayre, Isolt Sayre's father, is from a family that was transitioning from being Irish-speaking to English-speaking. In my opinion, William is probably the grandchild of the couple that made the change to speak English, which means that he probably cannot speak Irish, or at least not well.
HERE'S THE CUTE BIT; when William and Ríonach came back from Hogwarts, decided to get married and name their house, Ríonach most likely came up with the name "Oirbheart Muirneach", and then William tried to say it like an English-speaker of Irish heritage would say it, and ended up saying something like "Ilvermorny". Ríonach, laughing at him most likely, decided to keep it that way, and that inside-family-joke eventually became the name of the most democratic school in the whole wizarding world.
I really like my headcanon. ;)