Well, overall, you'd still as a group spend the least amount of time going for both at once, but if you insist on doing only Chloris during this time.
Mnk + mage on pop item duty. After you buy all the pop items you can on the AH (unless you just want to punish yourself and farm them all), mnk + mage will go around and mobs/NMs for the 3 pop NMs that drop key items.
While this is going on, war + whm will go around and kill NMs for Key items. War gets 10/13 of the WS' for red !!, so this is your combo for the NMs. Proc red and if red doesn't proc, deaggro the NM (if nobody else is around of course, and use conflux to do it) then reclaim to reset the trigger. This will take less time than farming a new pop set.
Check Chukwa from time to time, whoever claims have war come trigger it.
Then once you have a few pops, go ahead and pop it.
In your case, I'd probably just do 2 pops at a time, so the mnk's team can focus on killing stuff for pop items instead of having to come to the war's terf and pop NMs so he can get KI.
So I'd say KIs for war + whm, chloris x2, repeat while mnk and mage are farming pop items during this time. (anything from bloodshot hecteye to exorcised skull)