I recently posted about what I discovered worked reliably on the 3 person BCNM40 in Fei'Yin (there's also a 6 person there too, that's why I stress the 3

).
And I also said there was a lot of bad advice out there from people who spout theory without actually trying.
At the risk of being hypocritical, I offer an untested theory
Hand-to-hand damage is really what hurts the skeleton. Monks are a good choice by virtue that they specialize in hand-to-hand combat. The non-monk damage dealers that have tried fighting the skeleton never had hand-to-hand combat skilled up enough, if at all. So they used other weapons, but they weren't as good. That's why I recommend a monk.
However...
Ninja and warriors,
could skill up in hand-to-hand
if they really wanted to.
If you used monk as a level 20 subjob, not only could you use that as a way to skill-up hand-to-hand halfway, but you could use the passive monk Martial Arts ability during the BCNM.
Once you've mastered hand-to-hand, up to something reasonable that a level 40 warrior or ninja could use (preferably a warrior), I honestly believe you could take the role of damage dealer in the BCNM. You'd still need a Paladin tank and a Red Mage healer.
Again, this isn't tested. But I don't see why it wouldn't work.