Topological insulators (TIs) with hexagonally warped Fermi surface are natural platforms for the nonlinear Hall effect, as warping breaks inversion symmetry while preserving time-reversal symmetry (TRS). Here we show that this inversion breaking alone is insufficient: although warping generates a strongly anisotropic Berry curvature, the preserved threefold rotational symmetry forces the equilibrium Berry curvature dipole (BCD) to vanish identically. We demonstrate that linearly polarized light removes this symmetry obstruction: in the off-resonant Floquet regime, it lowers the rotational symmetry while preserving TRS, thereby generating a finite BCD whose magnitude, orientation, and sign are continuously tunable by the light intensity and polarization. For realistic Bi₂Te₃ parameters, we show that the induced BCD reaches ∼0.03 nm, yielding microampere-scale nonlinear Hall currents under experimentally accessible conditions. Our results therefore establish Floquet symmetry engineering as a route to activating the symmetry-forbidden nonlinear transport on TI surfaces without breaking the TRS.