If you want to shoot 5.56, a 223 Remington chamber is probably a bad idea. The differences in the chamber dimensions causes higher chamber pressure in the 223 chamber, and 5.56 is a higher pressure cartridge than 223. Depending on the action and barrel diameter at the chamber it might be a non- issue, but I wouldn't count on it.
5.56 chambers are generally less accurate. They're a typically a little looser all around, and have a different throat. The ammo, the rest of the rifle, and the intended use of the rifle make that irrelevant in a lot of cases, but for a precision bolt gun the difference may be noticable.
223 Wylde is a compromise between the two. It takes some dimensions from one chamber or the other, and splits the difference on others. It's safe to shoot 5.56 ammo in a 223 Wylde chamber, but a 223 Wylde chamber is a little tighter than a 5.56 chamber and has a throat similar to a 223 Remington chamber.
Given the situation you described a Wylde barrel would be the best fit. Personally, I'd go with a 223 Remington chamber for a precision bolt gun and accept that I won't be able to use 5.56 ammo. I've had good luck with Wylde barrels on my precision ARs, but I'd still opt for a tighter chamber if I was chasing precision.
