The Batcave reads “Other players cannot play Surprises during your turn.” Which comes first? The canceling of the play (so its special power never comes into effect) or the play itself (preventing the Surprise)?
A: So, the idea behind the Batcave is that if you have it, it’s as though you’re in it, and it’s fortress-like power of über-surveillance means that you cannot be Surprised. Since the out-of-turn function of Surprises is to literally surprise the player they’re affecting, the owner of the Batcave is immune from the in-game version of being literally being surprised: other people using Surprises against them on their turn.
Furthermore, (and perhaps most pertinent to this question) the Batcave is special, and it’s power is simply invoked faster than any Surprise that could normally be played for it’s “surprising” (i.e. out-of-turn) function. In this sense it is “even more instantaneous” than Surprises.