If information is conserved and the universe is finite, it's also cyclic, and the entropy law is false.
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The second law: dS≥0 is not violated when entropy is conserved. If the system is coupled to an environment the LVN equation becomes the Lindblad equation, which does not conserve entropy in general, and typically increases the entropy of the subsystem.
The second law is that the entropy of the universe increases in the course of any spontaneous change (Peter Atkins, "The Laws of Thermodynamics", p. 49). So you must be arguing that nothing ever changes, making the law vacuous (and why isn't it dS=0, then?).
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theories of physics are relevant within a certain window of scale/energy. The second law is useful to describe any system other than the global wavefunction of the multiverse. It doesn't mean that it's irrelevant at all.
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Furthermore, technically the observable universe (any system of interest to humans is within this subsystem) is a subsystem of the universe, due to the Hubble horizon. That makes the observable universe an open quantum system, not closed, and its entropy increases over time.
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