I am not sure why you are asking, if you have the answer.
Now the long answer - Think about it, the body of the 223 is .375" in diameter the bore of the Blackout is .308. .308 does not clean up the shoulder of a .375" case, and the 221 reamer will not be long enough to clean it up either. With a bolt gun, you could set back a .308 barrel by trimming the chamber end off some and cut a new chamber, re-thread and then re=head space. With a Handi rifle that would not be possible. You could "stub" a .308 barrel in and cut a new chamber for the Blackout case.
Good Shooting and Good Luck