*Yes, people are so sick of "but I'm being ironic" there's Tumblr memes about it.
I disagree with the assertion in the linked blog post that it's not possible to satirize -isms - in fact, I firmly believe it's necessary to expose the utter ridiculousness of these beliefs. Which, incidentally, is what the OP did with her anti-bot.
As hipster racism has become more widespread, it’s also crept into more general society. Racist content appears in films and television shows, disguised as “satire,” it’s on the cover of major magazines, it’s in the pages of respectable newspapers. While explicit racism is viewed as socially unacceptable, racism disguised as irony or satire is evidently perfectly acceptable, especially if it comes from middle class white people with trust funds.
In fact, it's an awful article. Full of conjecture, strawmen and self-righteousness.
As for the bot, the ironic part is the creation and publication of it, not in the actual functionality. Sexism aside, I can't comprehend why someone would want zero-value-added noise added to a professional communications channel (I don't consider banter and joking zero value in this regard). Especially since this invited further noise, and now they're locked in: it's going to be hard to go and say "enough fun for now" and turn both bots of now without appearing sexist.
They are basically providing documented proof in writing of a hostile work environment. Sure, this employee handled it well and had a little fun with it, but there are many more litigious employees that will take your company to the cleaners with just a couple printed chat logs.
The fact that she's complained about it and her concerns were shrugged by male employees and supervisors makes it even more egregious. This kind of story makes ambulance chasers salivate...