Can a single, larger black hole be split into multiple smaller black holes?
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?
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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.
The thermodynamic argument
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.
The geometric/relativistic argument
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.
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