Wait a second, if it has a .223 chamber, it shouldn't be firing 5.56 NATO. If this practice were generally OK, there would be no market for the Wylde chamber; if CZ chambers had 5.56 tolerances, it would be darn weird for a Varmint gun, and would possibly effect accuracy, in addition to the hazards involved. Commercial .223 is certainly okay in a 5.56 NATO chamber, but supposedly not vice versa--the relationship is reversed in the case of .308 and 7.62 NATO. Customer service might think it's okay, but they, like the guys behind the counter, can't always be trusted to know what is and isn't safe.
Then again, though, you should probably take the word of someone in the reloading forum instead of mine.
Have fun with that rifle, by the way!