Bucksnort1,
The 5.56 and 223 Remington had identical peak pressures when they were introduced, at 52,000 units of pressure by copper crusher. The SS109 developed in Belgium and sent to us measured 6% higher pressure on our style of copper crusher, and when we adopted that round as M855, the 6% higher pressure value was retained. On European equipment, as near as I've been able to learn (still looking into it), the two rounds measure the same pressure, so this difference is an instrumentation artifact. European NATO and 223 ammunition is all loaded to the same pressure as measured on their channel transducer gear, AFAIK. We have stayed with the difference and M855 is loaded to 58,000 psi on the conformal transducer, presumably because the apparent difference carries in it.
In any event, keep in mind maximum proof pressure for your rifle is 148% above the usually given maximum pressure, and your gun has to tolerate that without damage (though not a steady diet of it). Also, that maximum pressure isn't a maximum in the usual sense of being an absolute, not-to-exceed limit. What is given as a maximum pressure in manuals is a number SAAMI calls the Maximum Average Pressure (MAP). It is an average result for a ten round string. Individual rounds within the string can go higher. SAAMI limits that with a spec called Maximum Extreme Variation (MEV) and in a worst case, that would allow one round out of the ten to be 118% of the MAP value. The European CIP limits individual rounds to 115% of MAP. Further, SAAMI allows the average peak pressure for ten rounds to increase as the ammunition lot ages and the bullets get stuck harder into their brass. That allowance is given a different name, being called the Maximum Probable Lot Mean (MPLM), and it is 106% over MAP in the 223. So if the ammo isn't very old, it should be within the MPLM and be perfectly fine to fire without undue wear. Even if it went over a little, as long as you don't detect sticky case extraction, you should be good to go.
I fully expect you will be fine. I've never heard of a case of gun damage traceable to shooting NATO ammo in a 223. The warning against doing it is more geared toward avoiding long military specialty ammo that could jam in the rifling lands due to the 223 standard chamber's slightly shorter freebore. That could raise pressure another 20% or so if it happened. If it really worries you, you can get a 5.56 chamber reamer run into your existing chamber just far enough to make it match.