You missed something: the gravitational waves.
A black hole merger spacetime contains gravitational waves leaving the merger at the speed of light. Time reversal reverses time across the entire spacetime, and this converts those escaping gravitational waves into a converging gravitational wave front, as well as the black holes into white holes. These waves converge in on the central (now-)white hole and get so strong at that central point of convergence as to be able to "buck" it apart into two separate white holes.
Without those incoming gravitational waves, such a split would not occur.
EDIT: As A.V.S. points out in the comments, in fact, a better answer to this question would be that the future evolution of white holes is in general undetermined, or better unrestricted, in the sense that multiple future trajectories from identical phase-space points will satisfy the dynamical equations, though of course that means still that we must highlight that a crucial element of the answer here is that the time reversal turns the black hole into a white hole. (Indeed, this is part of why they're called "white" - technically that's understating it: they can literally spit out anything - even unicorns, no seriously, it'd be entirely [though unlikely] consistent with the equations for a 1-horned ungulate to pop out, as much as literally anything else.)
In a realistic black hole collision case, which is what I assumed in the answer above, then of course, yes, you will have the gravitational waves and so forth and you do have to take them into account in the reversal. But the situation is even more serious.
Since the future evolution of white hole is unrestricted, you can build scenarios with a totally causeless, spontaneous split of the white hole, and have it be consistent with the dynamical equations. As it is a consistent evolution, it doesn't violate time reversal symmetry. The reason that the Universe isn't covered with tiny white holes is that they are next to impossible to form in the first place - and likely, general relativity is not the final description of these things.
(I want to point out that there is actually an analogy for this within ordinary Newtonian mechanics called "Norton's dome". It is not physically achievable, but is still a system within the mathematical theory which has a similar property of its present state being equiconsistent with multiple future evolution trajectories.)
The short answer to your question is that positrons are not really electrons moving backward in time, and the premise of your argument doesn't work. However, something like what you are saying, is responsible for Hawking radiation.
Slightly longer...
There are a set of words you can give when you do QED in flat space along the lines of "positrons are electrons moving backward in time", but you really shouldn't take these words too seriously. Even ignoring gravity, electrons can be converted into neutrinos and quarks when you include the weak interactions (beta decay, inverse beta decay), so the notion that there is only one electron in the world that is jittering backward and forward in time every time a photon is emitted just doesn't work. The closest you can get is the CPT theorem... since quantum field theory is invariant under reversal of charge, parity, and time, then time reversal (T) is equivalent to reversing charge and parity (CP), which formally exchanges particles with antiparticles. But there's no way to actually implement a T transformation in reality. In controlled conditions, like a particle accelerator, you can set up an experiment done with one set of particles and the same experiment done with the CP-transformed particles to see what happens, and mathematically the results will be the same as if you had applied a T transformation, but at no point has time actually been reversed.
Now, there is something funny going on quantum mechanically with the fact that timelike and spacelike directions are switched beyond the event horizon of a black hole. But since we are sophisticated enough to realize that positrons are not electrons flowing back in time, we know it's not quite as simple as saying positrons will flow out of the event horizon. If you work through the math, you will find the implication of the horizon is that there are modes which have a negative frequency with respect to an observer at infinity. Performing a Bogoliubov transformation, this means that the state that looks like a vacuum to an observer near the event horizon, will look like it has particles to an observer at infinity. This is in fact Hawking radiation, and (in very crude terms) this is the strategy Hawking used to discover Hawking radiation in his original paper on the subject.
If you are really insistent, you can use these words to very crudely describe Hawking radiation: "particle-antiparticle pairs pop into and out of existence in the quantum vacuum, and near the event horizon sometimes a particle will escape the black hole and an anti-particle will fall in, or vice versa." You can loosely map these words onto the positive and negative frequency modes. But, like with "positron = electron moving backward in time" or "Feynman diagrams show trajectories of particles in spacetime," I would treat this more as a colorful analogy, than a rigorous description of what the math really says is happening.
As pointed out in the comments by @ChiralAnamoly, while I've phrased the answer in terms of the role of a timelike and spacelike coordinate switching roles at the horizon, physics cannot depend on your choice of coordinates. The coordinate picture in my answer is (I would argue) a fairly intuitive way of understanding what is weird about a black hole horizon and why you get negative frequency modes near the horizon, leading to Hawking radiation, it can be misleading to rely too much on coordinates. A more abstract but also more invariant way to describe what is going on, is in terms of different quantum vacuums states. An observer at asymptotic infinity will identify a certain state that is the "natural" vacuum, given the observer's worldline. An observer near the horizon will also identify a natural vacuum state. However, these two vacuum states are not the same. Positive frequency modes with respect to the horizon-observer's vacuum state, will be a mix of positive and negative frequency modes with respect to the asymptotic observer's vacuum state. This mixing gives rise to particle creation, aka Hawking radiation. The vacuum states for each observer are coordinate-invariant, as is the statement about the mixing positive and negative frequency modes (although the construction of the state is most easily done using coordinates adapted to the time of each observer).
Best Answer
In classical black holes, there's a time asymmetry: the event horizon can be crossed from outside to inside only, while if you reverse the arrow of time then it can be crossed from inside to outside only. White holes are by definition black holes with the arrow of time reversed. The proof that black holes can merge but not split also proves that white holes can split but not merge, and all other time-asymmetric properties of black holes are also reversed in white holes.
In quantum gravity, it should be possible for a black hole to split by emitting a black hole as Hawking radiation, though it's vanishingly unlikely (as unlikely as any other process involving a similar decrease in entropy). It's not clear that quantum black holes have any properties that are asymmetric in time for any reason other than ordinary thermodynamic irreversibility. You can still consider a black hole that is time-reversed in the same sense that a positron is a time-reversed electron, but positrons don't have a reversed thermodynamic arrow of time, and there's no obvious reason why quantum black holes would have one either. So there may be no distinction between black and white holes in quantum gravity.