We present a regularization approach to the study of the quantum dynamics of the mixmaster universe that allows us to approximate the anisotropy potential with the explicitly integrable periodic three-particle Toda potential. This approach is based on a Gaussian weighted Weyl integral quantization. Such a procedure naturally amplifies the dynamical role of the underlying Toda system by smoothing out the three canyons of the anisotropy potential. Since the respective eigenfunctions can be explicitly constructed, our finding paves the way to a perturbative approach to the quantum mixmaster dynamics.