Accurate modeling and control of transient thermal fields are essential for safe and effective focused ultrasound neuromodulation, where therapeutic efficacy must be achieved without inducing irreversible tissue damage. Classical Fourier-based bioheat models and forward simulation strategies are inadequate for this purpose due to their inability to capture finite thermal propagation effects and to enforce strict thermal-dose constraints. In this work, a novel inverse computational framework is developed for thermal-dose-controlled ultrasound neuromodulation based on a generalized non-Fourier bioheat model and a Laplace-domain Boundary Element Method (BEM). The governing bioheat equation incorporates phase-lag and thermal memory effects to account for rapid, localized energy deposition in layered brain tissue. By applying the Laplace transform, the transient non-Fourier bioheat equation is reduced to a Helmholtz-type boundary value problem, enabling an efficient and highly accurate boundary integral formulation. A dual reciprocity strategy is employed to treat domain heat sources, while continuity conditions are rigorously enforced at tissue interfaces. An inverse optimization procedure is then introduced to identify time-dependent ultrasound power amplitudes that produce a prescribed neuromodulatory thermal dose window while satisfying strict temperature safety constraints. The proposed framework is validated through analytical benchmarking, cross-comparison with finite element solutions, boundary mesh convergence studies, and inverse reconstruction accuracy assessments. Numerical results demonstrate that the method achieves precise thermal dose control, stable temperature evolution, and physically realistic control profiles with significantly reduced computational cost compared to volumetric methods. The developed approach provides a powerful computational tool for the inverse design and optimization of noninvasive ultrasound neuromodulation protocols and establishes a new class of boundary-based inverse bioheat transfer models with broad applicability in biomedical engineering.