Interface consistency and error convergence are central issues in concurrent multiscale computational methods, particularly critical for atomistic-to-continuum coupling models. However, existing theoretical studies remain limited and are mostly confined to one-dimensional settings. This work focuses on the multiresolution molecular mechanics (MMM) approach and systematically investigates the impact of various energy sampling schemes on interface consistency and error convergence. Two-dimensional square and triangular lattice models containing both atomistic and coarse-grained regions are constructed under bilinear element interpolation. The results show that interface secondary sampling schemes can significantly improve consistency in the interfacial region, with the scheme incorporating all neighboring layers achieving the best performance. Error analysis reveals that discretization error dominates the total error, and increasing the number of secondary sampling points effectively reduces the sampling error, particularly under tensile loading conditions. Moreover, both lattice types exhibit consistent error convergence behavior, demonstrating high generality of the method to different structures. This study highlights the advantages of energy sampling strategies in improving interface treatment and convergence behavior in MMM, providing theoretical support for the development of high-accuracy multiscale computational mechanics methods.