We can ask the same question without mixing in Hawking radiation and a pair of particles. If a particle falls into the black hole, it can't come out. And time reversibility won't get it out of there either. In fact, it is not a time reversible process. The reason for this is explained in A. Zee's book “Einstein Gravity in a Nutshell” (pp. 416--417). I'll just quote it:
A common confusion about plunging into a black hole
Confusio speaks up: “I have learned that the fundamental laws in classical physics (and also quantum physics) are time reversal invariant, that is, they are unchanged upon $t\rightarrow-t$. I read that if we take a movie depicting a microscopic process and run it backward, the reversed process must also be allowed by the laws of physics. So why can’t I run the film of the observer radially plunging into a black hole and watch him come flying out?”
Well, well, that Confusio is more astute than we think. Indeed, the Lagrangian
governing the motion of a particle in Schwarzschild spacetime is manifestly invariant
under $t\rightarrow-t$. So where is the catch in the standard arguments about time reversal invariance?
The catch, as I have already mentioned, is that the coordinate time $t$ increases to $+\infty$ as $r\rightarrow r^+_S$ and then decreases from $+\infty$ after the observer crosses the horizon. Indeed, as is evident from the Lagrangian just displayed, $t$ and $r$ exchange roles for $r<r_S$. The letter “$t$” no longer denotes time! Much more on this in the next chapter.
The standard arguments about time reversal invariance work perfectly well as long as $r>r_S$. Thus, if we could somehow install a trampoline at $r^+_S$ just outside the black hole, the observer in radial plunge could bounce back out to $r=\infty$, retracing his trajectory.