Thanks for all the great information in this thread.
Based on feedback:
1) You don't have to have the Ritual of Doom order hall trait for your order hall, just kill the demons (multiple sources)
2) You don't have to be level 110 according to Gummiberg who got it at lvl 106. That means a certain level of Artifact Knowledge and faction rep is probably not required.
3) Several people have stated that you can get the drop on any kill during the day, not just the first. Not sure we have enough consensus to say this is 100% true, but no reason at this point to doubt it.
So it seems it is just a random drop with no AR or Rep gating. So the best approach is to just join as many Ritual groups as you can per day to grind it out.
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