Although the t-z load-transfer method remains a cornerstone in geotechnical engineering, reliance on standardized empirical curves frequently introduces significant settlement prediction errors. To bridge this critical performance gap, this study develops an objective, automated back-analysis framework leveraging the Generalized Reduced Gradient (GRG) Nonlinear optimization algorithm (Excel Solver) to extract site-specific t-z and q-z parameters directly from full-scale static load tests. Integrating a one-dimensional finite difference model, the research evaluates the axial behavior of large-diameter bored piles by systematically comparing classical Bilinear (elastic-perfectly plastic) and continuous nonlinear Hyperbolic formulations against two rigorous benchmark datasets: Shibuya et al. (dense sand) and O'Neill et al. (cohesive clay). The results demonstrate that the Hyperbolic model markedly outperforms the Bilinear approach by gracefully capturing progressive soil stiffness degradation. Quantitatively, the Hyperbolic formulation achieves exceptional predictive accuracy with Pearson correlation coefficients exceeding 0.997 and dramatically lower Root Mean Square Errors (RMSE) across both granular (1.234 MN for Shibuya) and cohesive (19.086 kN for O'Neill) strata, whereas the Bilinear model exhibits severe error escalation (3.386 MN and 74.268 kN, respectively) despite maintaining relatively high correlation ( ). Furthermore, comprehensive sensitivity analyses and Monte Carlo simulations (N=300) reveal that pile-head settlement is predominantly governed by initial stiffness mobilization rather than ultimate shear resistance, with sensitivity ratios reaching up to 30,000. By rigorously integrating automated optimization, statistical validation (t-tests), and probabilistic uncertainty quantification (95% confidence intervals), this study delivers a robust, efficient, and reliable methodology that significantly enhances the predictive confidence of modern pile foundation design.