My op has Irresistible Faefolk in base, a troop on Battlefield A, and I have a unit with a hidden unit at Battlefield B. On op's turn, he moves his Bunny to his controlled battlefield and pulls my troop. Can I, in response to his Bunny moving my unit, play Last Caress and reveal my hidden troop, buff it, and play it to op's battlefield?

Yes — that works, as long as you let Last Caress resolve before you play the hidden troop.

Last Caress is [Reaction], so you can cast it while Irresistible Faefolk's move trigger sits on the chain, and it explicitly covers "the next unit you play this turn, even from face down." Normally Rule 811.1.d.1 forces a hidden permanent to be played to the battlefield it was hidden at, but Last Caress overrides that — the Core Rules example at 811.1.d.3 is exactly this: play a spell granting "the next unit you play this turn can be played to any battlefield," then play a hidden card to a different battlefield. Your troop also enters already buffed (+1 [M]) from Last Caress, and if it already has a buff it just doesn't get a second one.

Ordering is the one thing to get right: play Last Caress, pass, let it resolve (empty effect), then play the troop from facedown to your opponent's battlefield — the hidden card has [Reaction] while facedown, so you can still do it in the closed state with the Faefolk trigger on the chain. If you put the troop on the chain on top of Last Caress before it resolves, the permission isn't active yet and that play is illegal.

Also note the Faefolk trigger locked its target when it was finalized, so this doesn't save your Battlefield A troop from being pulled — and once your troop is sitting at a battlefield your opponent controls, that battlefield is contested, so a Combat Showdown comes after the chain empties.