Any cursory search of literature will find this term written as both bated and baited. The question becomes which is correct. (Now what Webster says is clearly known but where would be the blog post worthy rant in just following that old worthy blindly?) While baited breath might mean a peculiar method of hunting, bated breath is something of a curiosity. What does bated even mean?
Fortunately there is an answer known thanks to the old Bard himself. In "The Merchant of Venice" Shakespeare uses the turn of phrase in a particularly vicious sarcastic monologue and form context it becomes clear that bated is a shortened version of abated, which meant and still means 'to bring down, lower or depress' so basically to hold one's breath.
The issue was clouded a handful of centuries later when one lesser poet, Geoffery Taylor, wrote "Cruel, Clever Cat" and described the pernicious beastie waiting with cheese baited breath.
Now this author is prone to take the words of the Bard more seriously than an early twentieth century children's poet...except that it is quite common for barn cats to eat the seeds left out for songbirds....
As yet another old, dead author said, "Curiouser, and curiouser."