We characterize the anisotropic differential ac-Stark shift for the Dy $626$ nm intercombination transition, induced in a far-detuned $1070$ nm optical dipole trap, and observe the existence of a magic polarization for which the polarizabilities of the ground and excited states are equal. From our measurements we extract both the scalar and tensorial components of the dynamic dipole polarizability for the excited state, $alpha_E^text{s} = 188 (12),alpha_text{0}$ and $alpha_E^text{t} = 34 (12),alpha_text{0}$, respectively, where $alpha_text{0}$ is the atomic unit for the electric polarizability. We also provide a theoretical model allowing us to predict the excited state polarizability and find qualitative agreement with our observations. Furthermore, we utilize our findings to optimize the efficiency of Doppler cooling of a trapped gas, by controlling the sign and magnitude of the inhomogeneous broadening of the optical transition. The resulting initial gain of the collisional rate allows us, after forced evaporation cooling, to produce a quasi-pure Bose-Einstein condensate of $^{162}$Dy with $3times 10^4$ atoms.
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