My understanding is that this chamber will handle the hotter rounds and larger projectiles of the mil-spec ammunition, as well as the standard .223 ammunition.
Yeah, isn't 5.56 stamped right on the side of the upper?
Very generally speaking, it's typically suggested not to fire 5.56 ammo, or at least not a bunch, in a rifle designated as caliber .223. However, a recent article I read, backed with pressure measurements, suggested the ammo spec difference certainly isn't the only consideration.
The individual chambers play in a lot too. While there is a 5.56 chamber spec and a .223 chamber spec (and variants on those like .223 Wylde), not all 5.56 chambers are the same and not all .223 chambers are the same. Depending the chamber tolerances, it's entirely possible you could fire a 5.56 round in a "loose" .223 chamber and have less chamber pressure than firing a .223 round in "tight" .223 chamber.
But, in your case, 5.56 is the correct ammo. There is little point to shoot .223 in it unless maybe you were hunting and wanted more of a hunting style/controlled expansion bullet. As for plinking, 5.56 is usually cheaper for the run of the mill 55gr FMJ than for .223. Also, hard to say if that would have the faster twist barrel to properly stabilize the 62gr or heavier bullets.