Sorry for the delay with my judgment. I had unexpected (but welcomed) company last night.
Anyway, here we go:
After hearing both sides of this argument, examining and re-examining the evidence, doing some background research of my own, considering the pre-existing precedents of prep areas and era specific draft choices, and thinking about what would be the best thing to do in the interest of fairness, I've come to this conclusion.
During the majority of Cogliostro's character history, which, since Id did not state an era specific version, would mean, by default, he gets the latest incarnation of Cog and the character's entire canon history, I have no evidence which would lead me to believe the crown is "standard equipment" for Cog, along the lines of Captain America's shield or Batman's batarangs. For most of his character history Cog did not use any powers at all due to having been a hellspawn, and the issue that if he'd used up the last small bit of his symbiote powers which he had left, he'd be condemned to hell. He only obtains the Crown and Throne after committing suicide to gain access to hell, and then tricking and setting a trap for Spawn utilizing an artifact given to him by Mammon. Then, with Spawn's relinquished crown and the vacant throne now in his possession due to the coup, Cog uses the powers of these newly obtained artifacts to restore Spawn's humanity and send him back to Earth, and then give control of hell to other demons so he could build his tower.
It is my opinion that this situation is more akin to Thanos or Adam Warlock obtaining the Infinity Gauntlet than it would be to Captain America or Batman's regular use of their Shield/batarangs. My ruling is that the Crown and Throne are in Cog's possession/he has access to them currently in Hell, but per his entire character history, these items are not his standard equipment, just currently in his possession/employ. The Throne and Crown could not be brought into combat by any means per the tech/weaponry rule, but using the powers of these artifacts in prep to enhance your team would be legal (per the previous Brainiac/Miracle Machine ruling).