I would like to point my opinion about discipline priest also. Things can repeat since we are looking for problems so dont tell me that someone already said that i can see it myself just wanna point from my perspective :) . Your disc priest is rly bad ^ (no offence pls) . Cant resist to comment on this..
1) Disc is running 2/4 t18. Means he need to keep his tier bonus buff up all the time for more effective healing. This is not how it should look like (https://www.warcraftlogs.com/reports/fqhtYJzMdar48CAQ#type=auras&fight=29&source=1&ability=186478). This buff must be at least 95% uptime. His penance usage is super low.
2) During 2 mins 0 times cast Power Word: Solace. Kinda silly tbh because this is his main Evangelism stack generator. And you need Evangelism for Archangel buff to get +25% increased healing and absorbing.
3) Also 0 Smite casts for Evangelism generation.
4) No Archangel used at all. At 2 mins mark he supposed to use archangel at least 2 times.
5) 4th problem also makes 5th one: After disc priest use Archangel they get buff that makes next Flash Heal or Prayer of Healing 100% crit. Passive skill called Divine Aegis makes all crit healing to applie extra shields. Since he dont use Archangel or Prayer of Healing his Divine Aegis healing is low.
6) Instead of healing ppl with penance when they get Befouled debuffs that need to be overhealed, hes healing with Flash Heal. Thats rly bad choice. Especially when hes running 2 T18 that makes his penance much more powerful than his Flash Heal.
7) His enchant on weapon is Shadowmoon (spirit proc). When you have that kind of gear with 2119 spirit you never go for this enchant its just waste of healing that he can have. Hist top 1 prio what he supposed to do before your next raid is to get mastery enchant on his weapon. That way his healing and absorbing will increase. He cant have mana problems at that much spirit from his gear. Also when he starts to use his Power Word: Solace on CD (main Evangelism and mana generator) he can drop all his spirit items and get mastery/crit gear.
8)His Glyph of Holy Fire isnt best choice for this fight also. Suggest him to run Glyph of Fade. That will help him to mitigate more dmg taken. Less dmg u take personally means less dmg taken over all group. That one of the keys for successful raiding. Another reason is that he rly dont need that extra range in this particular fight.
9) He does pre-shielding before the pull but he dont use intellect potion. Should do that 2.
10) https://www.warcraftlogs.com/reports/fqhtYJzMdar48CAQ#fight=29&type=auras&spells=debuffs&source=8&pins=0%24Separate%24%23244F4B%24auras-gained%240%240.0.0.Any%240.0.0.Any%24true%2468373373.0.0.Paladin%24false%246788%5E0%24Separate%24%23909049%24auras-gained%240%240.0.0.Any%240.0.0.Any%24true%2421786717.0.0.Druid%24false%246788 this shows that both tanks have bit low Weakened Soul debuff uptime, that means that disc not casting enough shields on tanks. Should cast his PW:Shield on tank as soon as debuff drops.
In conclusion what he supposed to do is pretty much what MrDLTE3 said but im going to repeat again: Pre-shield and pre-pot before pull. At pull cast Smite and Power Word: Solace till he gets 5 evangelism stacks Cast Prayer of healing to give Divine aegis for ppl (mostly for soaking groups). During Archangel spam Power Word: Shield, and use Power Word: Solace on CD. Also Use penance once in order to keep Reparation buff up. After Archangel drops he has several seconds to cast Smite few times to make sure he has 5 Evangelism stacks rdy for next Archangel. Also dont foget Power Word: Solace and Penance on CD. First cast Penance on Befouled ppl only then use Flash Heal. More Power Word: Shields on tanks and ppl who are soaking pillars.
Hope your disc priest will read this and fix his problems in order to become more useful for raid group and increase his performance in other fights not only this one. GL with progression :)