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The study of collisionless shocks and their role in cosmic ray acceleration has gained importance through observations and simulations, driving interest in reproducing these conditions in laboratory experiments using high-power lasers. In this work, we examine the role of three-dimensional (3D) effects in ion acceleration in quasi-perpendicular shocks under laboratory-relevant conditions. Using hybrid particle-in-cell (PIC) simulations (kinetic ions and fluid electrons), we explore how the Alfvénic and sonic Mach numbers, along with plasma beta, influence ion energization, unlocked only in 3D, and establish scaling criteria for when conducting 3D simulations is necessary. Our results show that efficient ion acceleration requires Alfvénic Mach numbers ≥25 and sonic Mach numbers ≥13, with plasma-β≤5. We theoretically found that, while two-dimensional (2D) simulations suffice for current laboratory-accessible shock conditions, 3D effects become crucial for shock velocities exceeding 1000 km/s and experiments sustaining the shock for at least 10 ns. We surveyed previous laboratory experiments on collisionless shocks and found that 3D effects are unimportant under those conditions, implying that one-dimensional and 2D simulations should be enough to model the accelerated ion spectra. However, we do find that the same experiments are realistically close to accessing the regime relevant to 3D effects, an exciting prospect for future laboratory efforts. We propose modifications to past experimental configurations to optimize and control 3D effects on ion acceleration. These proposed experiments could be used to benchmark plasma astrophysics kinetic codes and/or employed as controllable sources of energetic particles.