Blue Mage.

They could play up the copying aspect more heavily than the learning aspect.
It would still be cool if they could collect monster abilities and equip them as cross-class skills but I'd also let them temporarily "learn" other party member abilities, like when a Bard is playing Foe's the Blue Mage can copy it and can pick up playing it when the BRD is done, or they take a stoneskin buff from a player that still has it and copy the effect onto the tank.
Maybe a Mimicry skill where you duplicate a party members last 2-3 actions, an ability to copy those 'damage up' buffs monsters apply onto yourself along with a spell to duplicate them onto the party.
Less of a focus on permanently learning what you're seeing and more copying and duplicating it in the heat of battle...

It's not really the traditional 'learning skills from monsters' Blue Mage, but I doubt we'll ever get that in FFXIV. Got to think outside the box .

I'm not sure how you'd handle their regular rotations though. I was hoping we'd get Musketeer as a class with a Blue Mage/Gun Mage style job for it.