<p>Due to the distinct combustion properties of hydrogen compared to conventional fossil fuels, novel burner designs are required. Additive manufacturing techniques enable the realization of complex geometries, which can be obtained through simulation-based design optimization. However, the considerable costs associated with high-fidelity CFD simulations of the governing physical processes in hydrogen burners limits their direct use in optimization. Multi-fidelity surrogate modeling provides an efficient alternative by combining inexpensive low-fidelity with accurate high-fidelity simulation results. We present a modular toolchain for simulation-based shape optimization of combustion devices that integrates (low-fidelity) RANS and (high-fidelity) LESs, which provide the basis for a multi-fidelity surrogate model. Inherently, this approach incorporates time-averaged and transient characteristics of the newly designed burners. Our approach is therefore capable of capturing both, quantities derived from averaged solutions and those influenced by transient dynamics. We apply the multi-fidelity toolchain to the optimization of the internal geometry of a weakly turbulent hydrogen burner nozzle. The results show that purely geometric modifications can significantly influence mixture characteristics through changes in flow features such as recirculation zones and jet-in-crossflow interactions. The optimization toolchain reliably produces designs that meet the desired mixture characteristics in both time-averaged and transient metrics. The results demonstrate the efficacy of this toolchain for the design optimization of burner nozzles (or other combustion devices) and it is suited to make use of the design freedom offered by additive manufacturing.</p>