If the laws of physics are time reversible, then I could start my problem with these two same particles with their momentums reversed, and the solution should be a black hole that splits into the two particles.
There is a really subtle, and in my opinion beautiful, detail in this statement: the definition of black hole is not time-reversible.
From the moment you say "black hole", you gave up on reversibility. By definition, a black hole is the region of spacetime which no observer that goes to infinity in infinite time can see (see this wonderful PBS Spacetime video for more details). This definition assumes that the black hole region can't be viewed from the future, and it is not symmetric with respect to time reversal. An analogue definition is that of a white hole.
This asymmetry in the very definition of black hole is what allows, for example, the result that a black hole's area can only increase over time, which explicitly distinguishes past from future. The answer to your question is then essentially the same: since the very definition of a black hole already distinguishes past and future, no, you can't find a black hole splitting into two particles by attempting to invoke time reversal symmetry.
Notice that this result that the area of a black hole always increases is fairly similar to the Second Law of Thermodynamics. In modern days, it in fact is interpreted as the application of the Second Law of Thermodynamics to systems involving black holes, as I discussed in this a bit more technical post.
