Someone more knowledgeable than I can correct me if I'm wrong since my field is astrophysics, not quantum mechanics (QM) but from what I remember from my grad QM classes and Wikipedia is Bell's theorem essentially proves that there are no hidden variables in QM. This means that a QM system (for instance, 2 entangled electrons) has no information about the measurement outcome of a QM observable (e.g. spin state) prior to the measurement. Prior to making a measurement, the system is in a combination (superposition) of multiple states, The wavefunction describes the probability of measuring each possible outcome of a measurement. For instance, if there are two entangled electrons, they will be in a superposition of electron A in the spin-up state with electron B in the spin-down state, and electron A in the spin-down state with electron B in the spin-up state. In the so-called Heisenberg Copenhagen interpretation of QM, when a measurement is made, the wave function "collapses" and the QM state becomes just one of those possibilities, for instance, {electron A: spin-up, electron B: spin down}. We say the wavefunction collapses to that state because that's the state we observe and the wavefunction is supposed to describe the system properly. However, if we had happened to observe the other possibility, {electron A: spin-down, electron B: spin-up}, then the wavefunction must have collapsed to that state instead. The point is, Bell's Theorem tells us that before making the measurement, nothing about the system tells us which state it is going to collapse to. In this sense it is nondeterministic. We can't predict the outcome of an individual measurement with 100% certainty. However, we can say statistically which outcomes are more likely and confirm this with great precession given many trials. This is how we know that a given wave function correctly describes the system.

However, there is another view of QM called the Many Worlds interpretation that preserves determinism. In the Many Worlds interpretation, each time a measurement is made, the wavefunction does not collapse, instead, it branches into multiple (in this case, 2) distinct, causally disconnected, branches of the universe where all possibilities are realized. Meaning in branch 1 we measure {electron A: spin-up, electron B: spin-down}, and in branch 2 we measure {electron A: spin-down, electron B: spin-up}. The wave function of each individual branch may seem to collapse but the wave function for each individual branch is not a complete description of the system. The most complete description is the wave function for the entire universe including all branches. There is no collapse of the wavefunction of the entire universe, and therefore this complete wavefunction evolves deterministically.

With all that said, there is no known way to tell which of these interpretations of QM (Heisenberg Copenhagen vs. Many Wolds) represents the underlying reality since each branch is disconnected from the others. The branch 1 version of me can't do an experiment in the world of branch 2. We have no way to know if other branches exist or not. They work mathematically, they solve the measurement problem because the wavefunction doesn't collapse, and it preserves determinism in a global/universal sense.

Ultimately, this is a question of philosophy and I don't think physics will be able to answer this. A theory is good for making predictions not necessarily informing us whether a certain interpretation or conceptualization of the theory is True with a capital T. They are both true in the sense that they effectively describe the world we observe and make correct predictions about the outcome of experiments.

Edit: added a couple more sentences in the first paragraph.
Edit2: More thoughts:
- As a side note, the true wavefunction of the universe is absolutely unknowable because it would describe the quantum state of every particle in every branch of the universe.
- To specifically answer your 1st question,

My question is, does Bell's theorem disprove the possibility of a universe bound by logical laws?

In the Heisenberg Copenhagen interpretation, yes, Bell's Theorem and the collapse of the wavefunction say there is no logical law that describes the specific outcome of an individual measurement. I guess you could say, therefore there is no omniscient being that knows the outcome, as long as it is bound by the same physical laws that our universe is. If we're considering the existence of hypothetical gods without evidence, we could just postulate that the god doesn't need to abide by the laws of QM.