Recent scanning tunneling microscopy experiments on graphene at charge neutrality under strong magnetic fields have uncovered a ground state characterized by Kekulé distortion (KD). In contrast, nonlocal spin and charge transport experiments in double-encapsulated graphene, which has a higher dielectric constant, have identified an antiferromagnetic (AF) ground state. We propose a mechanism to reconcile these conflicting observations by showing that Landau-level mixing can drive a transition from AF to KD with the reduction of the dielectric screening. Our conclusion is drawn from studying the effect of Landau-level mixing on the lattice-scale, valley-dependent interactions to leading order in graphene's fine structure constant κ=e^{2}/(ℏv_{F}ε). This analysis provides three key insights: (1) valley-dependent interactions remain predominantly short-range with the m=0 Haldane pseudopotential being at least an order of magnitude greater than the others, affirming the validity of delta-function approximation for these interactions. (2) The phase transition between the AF and KD states is driven by the microscopic process in the double-exchange Feynman diagram. (3) The magnitudes of the coupling constants are significantly boosted by remote Landau levels. Our model also provides a theoretical basis for numerical studies of fractional quantum Hall states in graphene.