We examine the near-field radiative heat transfer between finite-thickness planar fused silica slabs covered with graphene gratings, through the utilization of the Fourier modal method augmented with local basis functions (FMM-LBF), with focus on the lateral shift effect. To do so, we propose and validate a minor modification of the FMM-LBF theory to account for the lateral shift. This approach goes far beyond the effective medium approximation because this latter cannot account for the lateral shift. We show that the heat flux can exhibit significant oscillations with the lateral shift and, at short separation, it can experience up to a 60%-70% reduction compared to the aligned case. Such a lateral shift effect is found to be sensitive to the geometric factor d/D (separation distance to grating period ratio). When d/D>1 (realized through large separation or small grating period), the two graphene gratings see each other as an effective whole rather than in detail, and thus the lateral shift effect on heat transfer becomes less important. Therefore, we can clearly distinguish two asymptotic regimes for radiative heat transfer: the LSE (Lateral Shift Effect) regime, where a significant lateral shift effect is observed, and the non-LSE regime, where this effect is negligible. Furthermore, regardless of the lateral shift, the radiative heat flux shows a non-monotonic dependence on the graphene chemical potential. That is, we can get an optimal radiative heat flux (peaking at about 0.3eV chemical potential) by in situ modulating the chemical potential. This work has the potential to unveil new avenues for harnessing the lateral shift effect on radiative heat transfer in graphene-based nanodevices.