To be fair (which I never am), the phrase bated breath is the only survival of the verb bate, although its sister abated is still putting it about. Bate is a fossil word, a sole and strange survivor from years of yore that still creeps amok in some necks of the linguistic forest. So confusing it with the still common bait is understandable, even if it sticks in a pedant's craw.
I've already posted on some other examples in Gormless, Feckless, Ruthless and Reckless.
You see that bag at the bottom of the neck? That's a craw. It's where birds store food before digesting it.
P.S. The origin of amok (a Malay habit of suddenly and inexplicably dashing about trying to kill people) is fascinating and there's more here.