"Irishism" would be a perfectly reasonable term for the promotion of Irish identity and homeland. Lean into all things Irish! Generally we make do with "Irish pride" and "Irish nationalism" and such instead. It's not such a huge deal these days that we need a single term.

A lot of the uses of "transgenderism" reference the idea that a transgender lifestyle is fine, and a lot more consider the lifestyle to be a choice (which transgenderism advocates for). The problem isn't that the term is used incorrectly! It's used in the proper way for an -ism, for the most part. The problem is that the beliefs surrounding it are wrong, at least in part. It is not a meaningful choice for many people: you could choose to try to hide your feelings, but not to have different feelings. Thus, "Beliefs supporting trans identity = transgenderism" isn't a problematic usage of the term (so it's not a "faux term"). The problem is the next step, where the relationship between "transgenderism" and being trans is considered (and also in what "transgenderism" actually entails--after all, liberalism is a perfectly fine term but people on the right routinely ascribe to it all kinds of things that actual liberals don't believe and aren't among the tenets or consequences of liberalism).

"You shouldn't use the term transgenderism because it's overwhelmingly used as part of a framework of false beliefs and propaganda against trans people" is quite a reasonable take. But it's not because "transgenderism" itself is a problematic term in the way you describe.

(Incidentally, I have seen good research that indicates that "not a choice" is clearly true in some cases; the converse, that there is essentially never a meaningful choice isn't something I've ever seen explored in a satisfying way--not even for sexual orientation, let alone trans identity.)