No one can effectively answer this question. Batman has never speculated or mused for our edification as to why his mask has a port for which bullets could strike him if he wasn't as capable of using misdirection and psychology to prevent it.
Batman's appearance is based on the psychological effect his profile has. Like other vigilantes of the era he was related to, like the Shadow, his profile was meant to make a striking and imposing image designed to cause fear in anyone who had a reason to be afraid.
However, since he was still supposedly on the side of the angels, he had to have at least part of his face visible like heroes who were considered on the up and up like Superman who refused to wear a mask.
The most likely compromise would be to expose the expression center of his mouth, allowing him to appear both potentially more threatening when needed, and less as the situation demanded.
Before his costume was redesigned by Bill Finger, this was the costume the Batman would have entered the world with as designed by Bob Kane. Fortunately for history, Bill Finger worked toward making the Batman more unique than other similar heroes of the period.
Bob Kane’s original sketch of the character was totally different from the Batman we know today. Kane showed the very first drawing of a character he had first named the Bat, then Bat-Man, to Bill Finger who was the writer he hired to write the first Batman stories. Bill thought that the character looked too much like Superman, so he suggested major changes that would prove to be everlasting to the character’s legacy.
Bill got a Webster's Dictionary off the shelf, looking for a drawing of a bat, and found one. He then said to Bob Kane, 'Notice the ears, why don't we duplicate the ears?' He then suggested that Bob Kane would draw what looked like a cowl, to bring the nosepiece down and make him look mysterious and not show any eyes at all. Finger didn't like the bird-like wings, so he also suggested to Kane to re-design them and make a cape instead, and scallop the edges so it would flow out behind Batman when he ran so it would look like bat wings. A pair of gloves were added, colored purple from the start but later changed to blue.
Batman's costume was a compromise between the mask of the Shadow which covered his mouth with his trademark scarf and distraction of his long coat, and the domino masks which were popular and worn by many "mystery men" of the era and eventually by even his sidekick, Robin. Seen in the light of comparison, Batman's cowl isn't all that peculiar, but a variation on a theme. There are at least six other cowl wearing heroes in the JSA.
The original Justice Society of Earth-2 and its wide assortment of "mystery men and women" of the 1940s and 1950s.
The answers for why he uses a mask which reveals his mouth are left to speculation including one of the most often mentioned:
His fighting techniques need a adequate air supply best served by leaving his mouth open. But then this doesn't explain his various helmets with the ability to close the mouth plate, such as when he fights in an atmosphere filled with gas, smoke or other breathing impediments.
I suspect it is a compromise between total utility (the ability to entirely protect himself from external harm) which he does utilize when he use his full suits of armor such as the Hellbat, and the ability to still emote effectively both in an effort to terrify or console as the need arises.
- Batman is a psychological fighter. He recognizes a need to use fear as part of his arsenal of weapons. As such, nothing works the the fear a man who is smiling while he is kicking your butt, to undermine the well armed but often cowardly lot who confront innocent citizens of Gotham.
Another reason he may keep his mouth exposed (despite being portrayed as having a complete poker face, able to hide his emotional state from anyone) is his occasional need to emote in a positive fashion as shown below. A complete face mask would have never had the emotional impact necessary to save this young man without potentially turning him into a criminal after a conflict with the Caped Crusader.
However, most importantly how in the world is he supposed to kiss Catwoman wearing a full face mask?