Is there enough room in the mag well for the extra length of the 223???
I know a fellow that reamed out a CZ from 222 to the 223 so he could shoot cheap surplus ammo and the cartridges were to long for the magazine.
Likely it isn't a problem with the BSA but I would check it out and make sure. As Guntech pointed out, you will still be limited to 55gr or lighter bullets. As for accuracy??? I have both cartridges built on Rem 700s and in all honesty, they are in identical condition inside and out. If you were looking at any of the targets shot with handloads each of them likes, you would be hard pressed to tell which rifle shot them.