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 (\text{Cr}_{0.12}\text{Bi}_{0.26}\text{Sb}_{0.62})_2\text{Te}_3. Measurements of the complex Kerr angle, \Theta_K, were performed as a function of photon energy in the range 0.8\text{ eV}<\hbar\omega<3.0\text{ eV}. We observed a peak in the real part of \Theta_K(\omega) and zero crossing in the imaginary part that we attribute to resonant interaction with a spin-orbit avoided crossing located \approx 1.6 eV above the Fermi energy. The resonant enhancement allows measurement of the temperature and magnetic field dependence of \Theta_K in the ultrathin film limit, d\geq2 quintuple layers. 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. Show Partial Abstract