Mata - doesn't matter
Raptor - single target cures are pretty much all that you need, so sch is fine
Eft - Regenga is a handy counter to the poison, and the rest is pretty easy to deal with
Peiste - AOE -nas, particularly paralyze, are the main issue. Accession + Divine Caress makes it a non-issue for a whm. Regenga should take care of most of the AOE damage, so you can single-target cure and save strats for -nas. (Also Stona, but you know exactly when that's going to happen; pretty minor issue.)
Kurma - Pretty much all single-target curing and erases, aside from Tremor (which should hopefully be stunned). If Tremor hits, things could get messy with keeping up strategems, but other than that shouldn't be an issue. As a bonus, you can enspell the DDs to help speed up the kill.
Tojil - Perp/AOE Phalanx and Regen V would actually make the curing on this fairly easy. Phalanx should keep people from killing themselves on spikes, if that gets past, and Regen should cover most of the mundane damage. As long as you're careful about strat usage (or hit Tabula Rasa when needed) I don't see any reason it couldn't work out.
Main issue for sch is when you need a large curaga. Accession+Cure IV would likely be two macros, which can slow you down, and not being quick on the cure can end up with several people dead instead of just one.
Even a whm needs a bit of practice to do well in there. Not having done an NM run as sch, I can't really comment on any of the subtler aspects that might come into play, but on the surface I don't see any reason it shouldn't work.
The only significant issue (and really, it's the real issue you're trying to address) is making sure that you're fully committed to being a good healer. Every single person needs to bring their A-game to whatever job they're playing.