We study the self-organization of bidisperse mixtures of hard spheres in two dimensions by simulating random sequential adsorption (RSA) of tethered hard disks that undergo limited Monte Carlo surface diffusion. The tethers place a control on the local entropy of the disks by constraining their movement within a specified distance from their original adsorption positions. By tuning the tether length, from zero (the pure RSA process) to infinity (near-equilibrium conditions), the kinetic pathway to monolayer formation can be varied. Previously [J. J. Gray et al., Phys. Rev. Lett. 85, 4430 (2000); Langmuir 17, 2317 (2001)], we generated nonequilibrium phase diagrams for size-monodisperse and size-polydisperse hard disks as a function of surface coverage, size distribution, and tether length to reveal the occurrence of hexagonal close-packed, hexatic, and disordered phases. Bidisperse hard disks potentially offer increasingly diverse phase diagrams, with the possible occurrence of spatially and compositionally organized superlattices. Geometric packing calculations anticipate the formation of close-packed lattices in two dimensions for particle size ratios sigma=R(S)/R(L)=0.53, 0.414, and 0.155. The simulations of these systems presented here, however, reveal that RSA kinetics frustrate superlattice ordering, even for infinite tethers. The calculated jamming limits fall well below the minimum surface coverages necessary for stable ordering, as determined by melting simulations.