The proposes a surrogate-assisted multi-objective optimization framework tailored to structural design problems for which high-fidelity numerical simulations are computationally prohibitive. The approach is demonstrated on an axisymmetric laminated composite shell with several variables, including geometry, ply orientations, and plywise material selection. The optimization aims to simultaneously widen a frequency gap around a known excitation and minimize material costs. The main contribution lies in a comparison of three neural-network-based fidelity strategies. Beyond a conventional high-fidelity-only surrogate, two multi-fidelity formulations are introduced. The first employs an auxiliary refinement network that learns to transform inexpensive, low-fidelity finite element responses into high-fidelity, enabling the training of a single evaluator with dramatically reduced reliance on costly simulations. The second strategy uses a cascaded ensemble in which one network emulates the low-fidelity model and a second network maps its output to pseudo-experimental targets, explicitly preserving the low-to-high fidelity structure. All optimization runs are performed on surrogate models, with no finite element calls during the evolutionary search; final candidate designs are verified a posteriori. To objectively assess optimization performance, the study introduces a normalized relative hypervolume indicator computed with respect to an envelope-based approximation of the true Pareto front, complemented by standard metrics. This evaluation protocol isolates the influence of fidelity strategy from architectural or algorithmic effects. The results show that multi-fidelity learning substantially improves Pareto-front quality while reducing high-fidelity data requirements compared to the single-fidelity baseline. The findings confirm that surrogate accuracy is en important driver of optimization success, and demonstrate that carefully designed multi-fidelity neural pipelines provide a practical and scalable solution for high-cost structural optimization problems. Acknowledgements: Financed by the Minister of Science and Higher Education Republic of Poland within the program “Regional Excellence Initiative”.