Can you get a permit in WA state to conceal and carry a realistic bb gun

There is no reason a regular concealed weapons permit would not cover a bb gun. In some counties/cities, air rifles and air guns are included within the definition of other types of guns which would be subject to the concealed carry laws. As a practical matter, I would be concerned that carrying such a concealed weapon might indicate a lot more firepower than a bb gun packs, thereby prompting a more dangerous response than might otherwise be justified if the bb gun is exposed. So, while I wouldn't recommend it, a concealed weapons permit should allow the concealed carry of a bb gun.