Maybe check your manual to see if it'll say what the "switch ammo" button is on your system?
On the PC, it's hardcoded as hotkey "2". When I have a 5.56 weapon equipped, and I have several types of 5.56 ammo (5.56, 5.56 HP, 5.56 AP, .223), each time I hit "2" while the gun is equipped, the ammo type will switch to the next one.
So, hopefully your game manual will say what the equivalent of "2" is on your controller.
Hope this helps.
Thats what I was trying to say, the gun doesn't actually say it takes the .223 round but it does. Maybe your hotkey button isn't working try this ok,
1. Equip any weapon that uses the .556 round.
2. Go into the pip boy and go to the ammo section and look through the list to see if .223 is highlighted.
3. See if you can equip the .223 ammunition in the ammo menu with the .556 rifle equipped.
Let me know what happens.