While it is known that the double-layer Bi2Sr2CaCu2O8+y (BSCCO) cuprate superconductor exhibits an incommensurate superlattice (IS), the effect of IS on the electronic structure remains elusive. The interlayer interaction produces a double-peaked van Hove singularity (VHS) in the density of states (called bilayer splitting) which is strongly changed by the incommensurate period of the superlattice controlled by the misfit strain between different atomic layers in BSCCO. Recently the wave vector q = ǫb (where b is the reciprocal lattice vector of the average lattice) of the superlattice in a high-quality single crystal has been measured with high precision [N. Poccia et al., Phys. Rev. Materials 4, 114007 (2020)]. This work reports the calculation of the VHS bilayer splitting in a quasi-commensurate phase with a rational number ǫ = 9/43 for the pure BSCCO crystal. We extend the calculation to the devil's staircase observed in the high entropy perovskite Bi2Sr2Ca1−xYxCu2O8+y (BSCYCO), where the wave vector ǫ = 9/η changes in the range 36 > η > 43 for 1 > x > 0 controlled by the lattice misfit strain. Our results are of high relevance where the chemical potential is tuned in the optimum and overdoped regime where the superconductor to metal transition takes place. The similarity of the complex VHS splitting due to the devil's staircase in mismatched CuO2 bilayer with VHS due to moiré lattice in strained twisted graphene. This makes mismatched CuO2 bilayer quite promising for constructing quantum devices with tuned physical characteristics.