Let's raise to the top a claim that Bumble makes: Simulation is modeling using computation. Both mathematics and logic are forms of computation, and in computer science, we are interested not just in the formal systems that encode computation, but the physical processes that undergird computation (SEP). You ask:
Is it possible to simulate a reality like ours using other logic than classical logic?
So the question is not can logic and math be used to simulate the physical universe (they are routinely in dynamical simulation for engineering), but can it do so completely perhaps leading off to the question of whether or not we can fully simulate our own physical reality. This is known as the simulation hypothesis. From WP:
he simulation hypothesis, as formulated by Nick Bostrom, is part of a long tradition of skeptical scenarios. When it was presented by Bostrom as not merely a philosophical speculation but an empirical claim with quantifiable probabilities, the hypothesis received criticism from some physicists, such as Sabine Hossenfelder who called it pseudoscience and cosmologist George F. R. Ellis, who stated that "[the hypothesis] is totally impracticable from a technical viewpoint" and that "Late-night pub discussion is not a viable theory." Versions of the hypothesis have also been featured in science fiction, appearing as a central plot device in many stories and films, such as The Matrix.
The question isn't is there a "best" logic capable of simulating physical reality. There are classical and non-classical logics and mathematical theories such as differential geometry that can be used to simulate just about anything you can think of. I've yet to see an argument that something that we observe as a physical phenomenon can't be at least somewhat accurately simulated. Rather, it's that phenomenon themselves are stochastic. How do you model something that is random?
In computer science, pseudorandom number generators (PNRGs) are used all the time. But they are not random. Therefore, one cannot simulate random phenomena with PNRGs completely accurately. Often, in strong cryptographic systems, physical phenomena are used to produce physical randomness which is mathematically secure in the same way as a Vernam cipher (the non-CSPNRG sense).
How do you simulate the randomness in the universe besides using the randomness of the universe? Is it even a meaningful question? And what of particles who obey wave functions? How do you simulate Heisenberg's uncertainty principle? Ultimately it may not matter.
Coming back full loop to the nature of physical computation, there are limits to the physical computers we build. This means, we have to ask about the limits of a computer for simulating the universe as we understand it. For instance, it is not tractable to construct an actual Turing machine, because such an abstraction presumes an infinite tape for computation. There are space and time constraints in computation and those are studied in complexity theory. There are constraints on computation itself and those are studied in computability theory. And to simulate the universe would ultimately requires more resources than the universe itself has. Consider how complicated and how much physical mass and energy a quantum computer requires to model a handful of q-bits. So consider this argument.
Every simulated particle in the simulated universe (ignoring the fact they're more accurately described as point-like disturbances in a field) would need to correspond to at least one particle in the physical universe as part of the simulator. Then, additional particles are required to handle the act of transforming the state of those physical particles from a prior state to a new state for simulation (in CS, we call those states pre- and post-conditions in descriptions like Hoare logic). But we've already exhausted our particles in correspondence with the simulation. Thus the entire universe cannot be simulated a priori.
But maybe some some subspace of real space can be simulated. That is what a video game does. But the question of simulation our physical reality doesn't reduce to a question of which logic, but rather, what resources to construct the simulator.