We report measurements of the polar Kerr effect, proportional to the out-of-plane component of the magnetization, in thin films of the magnetically doped topological insulator (Cr0.12Bi0.26Sb0.62)2Te3. Measurements of the complex Kerr angle ΘK were performed as a function of photon energy in the range 0.8eV< ω<3.0eV. We observed a peak in the real part of ΘK(ω) and zero crossing in the imaginary part that we attribute to a resonant interaction with a spin-orbit avoided crossing located ≈1.6 eV above the Fermi energy. The resonant enhancement allows measurement of the temperature and magnetic field dependence of ΘK in the ultrathin film limit, d≥2 quintuple layers (QL). We find a sharp transition to zero remanent magnetization at 6 K for d<8 QL, consistent with theories of the dependence of impurity spin interactions on film thickness and their location relative to topological insulator surfaces.