In this work, we present a novel data-driven tuning framework for a class of nonlinear controllers, namely those based on the so-called hybrid integrator-gain system (HIGS). In particular, we focus on minimizing the settling time in point-to-point tasks, i.e., the time required for the error to converge and settle within a desired error bound after the task has finished. The proposed approach is based on sampled-data extremum-seeking control and allows simultaneous tuning of both linear and nonlinear parts of the controller, while guaranteeing input-to-state stability based solely on non-parametric frequency-response function data of the plant. These stability properties are guaranteed by a newly developed procedure for the data-driven verification of existing stability criteria. The efficacy of the proposed approach in tuning HIGS-based controllers for improving the settling time is validated extensively with a case study on an industrial wire bonder showing significant improvements in the worst-case settling time compared to LTI control.