Phrases such as “batshit nuts”, “batshit crazy”, “batshit insane”, “shit nuts”, etc, to describe someone as completely off their rocker.
Where do they come from? The closest thing I could think of to a factual answer would be getting rabies from actual bat feces, leading to strange behavior or insanity.
I don’t know. If I had to guess I would suppose that is somehow related to the old ‘Batman’ TV series. Dick Grays, Bruce Wayne’s lover (I mean ‘ward’) was always saying ‘Holy Bat (something), Batman!’
I think it’s just an amusing construction. Apeshit is used the same way. Doubt it could be traced to a specific origin. Robin said “holy this” and “holy that” but not “holy bat this-or-that.” All of Batman’s stuff was that Batmobile, Bat Cave, Bat Plane, same Bat Time, same Bat Channel. But I think the phrase predates the popularity of Batman as brought by the TV show.
Everything on the show was Bat-This and Bat-That. Robin kept saying Holy This and Holy That. It is no wonder that as the show has been off the air for almost forty years that the to meld together.