Horses and bulls make a bit of sense, big animals make big poos. Calling something a big poo means it is very stinky, a huge waste, etc. Very yucky. But bat poos? Those are teeny tiny, small bats make small poos. What do people have against the poos of bats?
Also, what about larger animals? Whale poos are huge, and yaks, buffalos, etc. all make big poos. Don’t ask me how I know.
That’s probably true regarding the dangers; but as far as I’ve ever seen, it doesn’t link to the etymology of it at all. Batty or similar things have a fairly established record. Batshit is less than a hundred years old as a term for crazy, and has solid origins in print that aren’t related to that.