If only you didn't have to waste precious segments to try, I bet more people would be willing to try different stuff or simply to help.
Yeah, everyone should just be able to try the fights infinite times, that would make the content so much more engaging after everyone beats every fight in a month and quits the game. You should design an MMO!
Im usually the blm on the attempts, but its always a missed sc window along with rayke and gambit that causes the time out.
This is indeed a problem but it sounds like it's a problem with the execution of your members and not a problem with your setup. This fight requires coordination and communication more than any other. While Rayke/Gambit/Subtle are up, people need to be focused on doing SCs and not get distracted by buffing, debuffing, doing RD, or whatever other nonsense. After those CDs go down, you can prepare for the next time.
Adding DRK, DNC, or whatever other job to make SCs seems nice, but who are you replacing? As others have said, COR RUN SCH BLM GEO are pretty much essential to "do magic damage to the boss" so your only real option is BRD. Do you think you'd rather have "more reliable SCs" over having ballad, march, etude, threnody, plus carol, march, madrigal, etc. on the tank? Imagine your RUN tanking with no haste support except embrava, -500 lightning resistance, and -250 accuracy; it seems to me that you're losing a SC partner to gain a SC partner.
I think it's worth considering putting BLM in melee range and you could try helping with SCs, but consider that you won't be able to close (or you'll miss out on MB time) so you'll only be able to contribute to the first part of a SC which is likely to be a Darkness, which is significantly worse than the 4-step you could otherwise be doing because of the MB bonus for longer SCs.
I would suggest getting your rhythm down with your existing group. Outside of RD/WC and dealing with the add, it should be incredibly easy for RUN and COR to do Upheaval -> Leaden -> Steel Cyclone -> Wildfire for 3 skillchains in a row. Between Samurai Roll, Earth Shot, Triple Shot, and capped haste on the RUN, TP flows very easily.
edit: oh and in case it wasn't obvious, we cut out SCH skillchains entirely in my groups' winning setups. Relying on a SCH slows things down in my experience as they have other things going on and will be distracted by buffing, curing, etc. whereas COR and RUN can just spam nonstop.