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Essentially, my question is, can one larger black hole be split into a system of smaller black holes?
I'm assuming energy requirements will be prohibitive, and that it may not be possible without exotic matter with negative mass. Can, if energy requirements are not a problem, and exotic matter is available, a black hole be split into multiple smaller black holes?
No, black holes are scientifically inseparable. No amount of energy can split them apart.
Black holes can "evaporate" with time, but this process is more akin to death rather than birth. If this deity can directly manipulate the fabric of spacetime, then splitting would become feasible. But if you want to explain it, you would have to create your own physics.
Alexander's answer is completely correct; there is no way to split one black hole into smaller ones. I think, though, that it might be worth explaining why this is the case, particularly because there are two independent lines of reasoning that come to the same conclusion. If you want to split a black hole in your universe, it turns out that you'd have to throw out both thermodynamics and geometry.
Black holes have entropy, which is related quite neatly to their areas: $$S=\frac{k_BA}{4\mathcal{l}_p^2}$$ where $k_B$ is Boltzmann's constant and $\mathcal{l}_p$ is the Planck length. Now, the second law of thermodynamics tells us that the entropy of an isolated system cannot decrease, and it turns out that we can generate inequalities telling us that if a black hole were to split into two smaller black holes and keep all the right quantities conserved (e.g. mass, charge, angular momentum), the entropy of the system would decrease. Therefore, such an event would violate the second law of thermodynamics, and therefore it is impossible.
There's actually a completely independent argument attributed to Hawking based purely on geometry, relativity and causality. The finer points of it are beyond me, but it boils down to the idea that splitting a black hole would cause two null geodesics to begin at the final point of contact between the two black holes, which turns out to be impossible.
We then have two independent arguments for why you can't split a black hole. Any worldbuilder interested in getting around them has to get rid of both thermodynamics and geometry (and, by extension, relativity) and still manage to create a self-consistent universe in which black holes can even exist. Ignoring one or more conservation laws might solve the thermodynamic objection, but then we're just being silly - and only addressing half of the problem.
While you can't split a black hole in a conventional sense I wouldn't completely rule this out because of Hawking radiation. The smaller the hole the hotter it radiates--but that's because more of the virtual particles are escaping.
What happens when you construct a handwavium machine that pulls all those virtual particles away from the black hole? It seems to me that you could turn the evaporation rate up as high as you have the energy to power it. Pulling these particles away will of course be incredibly energy intensive--but what if the machine is powered by dropping them down another black hole? I would not want to rule out the possibility of a machine that basically sucked mass out of one black hole and deposited it into another very close by.
Really, we have no idea what happens in a black hole because light cannot get out.
So just say they can with weird future tech, that uses their structure, and you will be fine.
I would go with:
They are made of substance X, that instantly explodes when ray Y is shone in causing the whole thing to explode in an outward expansion of spacetime, which allows the two halves to escape each other.
Also make them alive ⚕️ to make them more interesting.
HDE 226868's answer is completely correct. But what if we took two black holes in the process of merging and added a third black hole?
The first two would be just below each others' point of no return but still circle around one another. Viewers from the outside would just see a black hole with a lengthened shape rotating very rapidly, using indirect methods like laser scanning.
Now a third black hole of similar proportions as the other two together flies by at speeds close to the speed of light (let's say they are all three meeting in the center of a large group of black holes, and have been accelerating towards that group for millions of years, with some 'lucky' gravitational slingshotting by other black holes on multiple occasions).
The third black hole flies just by the two other black holes, with their event horizons overlapping slightly.
This would be a little bit like an unstoppable train hitting an inmovable wall, so the maths are a little beyond me. But I think we might not only get a gravitational slingshot effect strong enough to separate the two formerly joined black holes, but also the creation of a completely new fourth black hole where the three black holes overlapped, with the weight which the three original black holes lost in the encounter (probably just a small fraction of the three bodies - small enough to turn into a neutron star or an exploding cloud of protons). Entropy would continue to increase normally, as the black holes would all be moving very slow now (with a good chance of merging eventually).
