Off the top of my head, I can’t think of a word in English that ends with “is” while being singular, only plurals and uncountables come to mind, so I can’t really follow the examples of other words. What makes it even weirder, I’m not sure how to pronounce Illinoises… Would it be as written, or as if an Illinois was pronounced by someone who has never encountered it before? Illinoi are also meh, since now plural looks as a singular and the other way round.
I think over the course of years it would eventually end up becoming Illinoises, no matter if the word came from a different language. Words tend to get assimilated like that. It sounds weird now cause, well, there are no two Illinois so practically no one ever used it in plural. I’d pronounce it literally as „noises“ as in noise. But I‘m no linguistic expert, heck I‘m not even a native English speaker. It‘s just my belief.