This paper investigates the problem of high-precision target localization in integrated sensing and communication (ISAC) systems, where the target is sensed via both a direct path and a reconfigurable intelligent surface (RIS)-assisted reflection path. We first develop a sequential matched-filter estimator to acquire coarse angular parameters, followed by a range recovery process based on subcarrier phase differences. Subsequently, we formulate the target localization problem as a non-linear least squares optimization, using the coarse estimates to initialize the target's position coordinates. To solve this efficiently, we introduce a fast iterative refinement algorithm tailored for RIS-aided ISAC environments. Recognizing that the signal model involves both linear path gains and non-linear geometric dependencies, we exploit the separable least-squares structure to decouple these parameters. Furthermore, we propose a modified Levenberg algorithm with an approximation strategy, which enables low-cost parameter updates without necessitating repeated evaluations of the full non-linear model. Simulation results show that the proposed refinement method achieves accuracy comparable to conventional approaches, while significantly reducing algorithmic complexity.