RE: "I've always wondered why a pistol firing that type of ammunition isn't considered a short barreled shotgun."
For the same reason that pistols that fire "rifle" cartridges (e.g. Thompson Encore, Remington XP100, etc.) aren't considered "short rifles".
Basically, the rule is that the firearm is considered to be whatever it was manufactured to be and can't be changed and it doesn't depend on the ammo. For instance a machine gun that has been converted to semi-auto only is still considered a machine gun. (there is an odd offshoot of this that concerns AP ammo, but that wasn't really your question).
It's kind of a strange rule, but it's the only feasable means that the BATFE has for administering these issues.
Does this rat shot do damage to the rifle?