This paper proposes a novel data-driven control framework that combines Koopman-based linear parameter-varying (LPV) surrogate models and an integral quadratic constraint (IQC)-based error characterization to achieve effective closed-loop guarantees for nonlinear systems. In particular, we employ extended dynamic mode decomposition (EDMD) to approximate nonlinear dynamics. The residual errors are characterized directly from the data using non-parametric IQC multipliers in the frequency domain, providing a tight data-driven uncertainty characterization. The framework adapts to both linear and bilinear forms by appropriately selecting the scheduling variable. Moreover, an IQC-based characterization of the scheduling parameter enables frequency-domain LPV controller design, ensuring robust stability and performance. An iterative algorithm optimizes both the IQC multipliers and the controller parameters, reducing conservatism and ensuring monotonic convergence of the robust performance index. Numerical simulations validate the proposed approach and demonstrate convergence to tight performance guarantees using a finite number of data trajectories.