Wind farms experience reduced power production and elevated structural loading due to wake interactions. Wake-mixing control techniques, which dynamically excite upstream turbine wakes to accelerate recovery, have demonstrated promising improvements in downstream power production but at the expense of increased fatigue loading. Identifying the optimal control settings and quantifying the resulting load implications remain challenging because these methods require high-fidelity simulations that capture both the dynamic actuation and the resulting turbulence. Moreover, existing load surrogate models do not incorporate wake-mixing control, largely because conventional engineering wake models are unable to reproduce periodic wake excitation. This study presents two complementary advances to improve the design of wake-mixing strategies using a limited number of large-eddy simulations (LES) and Gaussian process (GP) regression. First, we develop an efficient simulation-driven framework to identify optimal frequency and amplitude parameters for wake-mixing control, yielding a clear optimal power gain of 7.5 % near a Strouhal number of 0.25 and pitch amplitudes of around 4° for a two-turbine array. Second, we present a surrogate model capable of predicting fatigue loads for wake-mixing control. Using LES-derived rotor-plane inflow fields for aeroelastic simulations, we construct a load database that encompasses various combinations of wake overlap, turbine spacing, and wind farm control settings. The result is a load surrogate model based on GP regression trained on sector-averaged inflow quantities that accurately predicts damage equivalent loads, including the effect of increased excitation in the wake. This model enables the joint evaluation of power gains and load penalties at the wind farm level, supporting a more informed design of wake-mixing control strategies.