Every job works, I know this for a fact, some set-ups may have a little bit more trouble then the other, Just a little. (Im not Neosquall, He does not have Chains of Promathia) Anyways... Lets go through a review of jobs and their roles in CoP:
WAR: Can Sub NIN, makes a pretty good tank early on, later on its too rough, still possible but rough, of course later on it becomes a Damage Dealer mainly.
MNK: Im not too Familer on MNK but I know they work as I've had a few MNK friends who've done these missions and it worked fine. I have no experience with MNK but I know people who used it.
WHM: Well...its pretty much a given if you want a healer in your group...yea...
BLM: Work on BC's like every job, but theres a few BC where BLM shine: Riverne Site B01, Bearclaw P., Mine Shaft #XXXX (What job doesn't work on those Moblins . (`_>`)

I have no experience with BLM but I know people who used it.
RDM: Pretty much the same as BLM, works on every BC, definatly a job choice starting at 50 Cap but it can be used before that. Has the same high points as BLM. I have no experience with RDM but I know people who used it.
THF: Yea, seems like it would not work on these BC no? Well it does, Also does very well on Bearclaw Snoll Tzar, Proof? Theres a video roaming around on FFXIonline, Allakhazam, etc.... I have no experience with THF but I know people who used it.
PLD: I've used PLD Tank for all the missions up to Omega/Ultima and will continue to use them, so on and so forth.
DRK: I personally do not know any Dark Knight who've done CoP as well...Dark Knight. I'm sure it can work fine if used right and I'm pretty sure someone , somewhere , uses/used DRK and it worked on these missions.
BST: Yea, while a very not so looked over job, its great for BC with Multiple Mobs since a Jugged Pet can hold for a bit, BST can also be a damage dealer w/|w/out Pet for the BC's. No experience with one.
BRD: Always nice to have mages get a refresh or the Melee do more damage to kill off the monster quicker right? Of course it is, back-up healing, great buffs, BRD definatly work while few do BRD. I know a few who have done BRD and did fine, but myself has yet to experience the use of one.
RNG: RNG work, no you do not need a RNG to do these missions, are RNG nice though? Yes... What do they do? Damage of course.
SAM: 60+ BC Battle, SAM mimics a RNG , quite well if I do say so myself..
piror 60, it still works just gotta find out the role to use it in.
NIN: Many people used NIN/WAR through out CoP, NIN is definatly a great tank, but theres some BC where its not the best, same goes for PLD, but they both work in each and every BC. As long as you have a tank.
DRG: See THF. In the video roaming around, they use a DRG, and....like I've said for every job posted above, it works for every BC.
SMN: The Last Job on the list, SMN Buffs (Stone + Blink) and 2hour are great, once again its not needed, but its always nice to have atleast 1 with you.
So as you see, each job has its high and low points, but they all work. Don't think just because you got one job at the level for the mission you can't do it. But you need a few things: Hope, knowing your job and role in the battle, and determination.
Please, if you have the "These missions are impossible, I give up!" or the "Im spending all my Gil on this, its not even worth it" or the "What a waste, worst expansion ever" Attitude, I suggest you try to fix that or do something about it, those are the exact attitudes which cause loss and you don't even have a sense of hope.
That is all~ 8)