The quantum theory of multiphoton-stimulated bremsstrahlung of charged carriers on an arbitrary electrostatic potential of impurity ion in doped bilayer graphene (BG) in the presence of coherent electromagnetic radiation is developed. A pump wave field is considered exactly, while the electrostatic potential of doped ions is considered a perturbation. The essentially nonlinear response of BG to a pump wave and the significant differences from the case of single-layer graphene, which can be associated with nonlinear parabolic dispersion, are shown. The latter opens a new way to manipulate the electronic transport properties of conductive electrons of BG by a coherent radiation field in the “domain from the terahertz to mid-infrared frequencies.”