The bore diameter of the 22LR and 223 barrels are slightly different. Also, 22LR barrels twist rates are usually 1/16 while the 223 barrels are 1/7 or 1/9. You have a few choices, but they are going to cost a bit. Surplus A1 1/12 barrels cost about $100-150 depending on condition. You won't get match accuracy, but it will work for both 22 and 223.
You can get the 1/12 barrel and have it cut and threaded to 16". The gas port will have to be plugged and the new one drilled. The front sight base will have to be moved.
I have an A1 barrel that was cut down to 11.5 and had a 5" flash hider permanently attached. I only wanted to shoot 22lr out of it so I didn't have the gas port drilled. For less than $100, KKF cut the barrel, threaded and recrowned it, and installed the FSB on it. I already had the barrel and wasn't going to use it, so that didn't cost me much. For the total value of the barrel and work, I guess I could have just bought a 22lr barrel and had it profiled to what I wanted. Oh well.
Or you can contact someone like
AR15Barrels.com and have them cut and thread a 1/12 barrel to 16". They can contour it to a light weight and have them install the gas port and FSB.
For the cost of the kit ($100-150), barrel+parts($125-200), and labor($100-200), you might just consider buying a 22LR upper from DPMS/Bushmaster, or settle for mediocre performance out of a 16" 223 barrel with a 1/9" twist.