The rapid advancement of machine learning and additive manufacturing has facilitated the development of architected materials with properties tailored beyond those found in nature. Inverse design provides a more data-efficient alternative to traditional trial-and-error strategies; however, most existing methods rely heavily on extensive datasets or limited high-fidelity simulation and experimental data. This challenge is particularly acute for architected materials exhibiting nonlinear mechanical behaviour, where accurately capturing complex deformation mechanisms demands costly evaluations. To address this limitation, a Multi-Fidelity Bayesian Optimisation (MFBO) framework is proposed for the inverse design of cellular composites, directly targeting their full nonlinear response. By combining information from models of different fidelity levels and quantifying response similarity through a scalar metric, the framework enables efficient design space exploration while reducing dependence on expensive high-fidelity analyses. As a proof of concept, the approach is applied to spinodoid cellular composites using finite element simulations, with validation from compression tests on short carbon-fibre reinforced PET-G specimens. Four target responses are investigated, and three multi-fidelity strategies are benchmarked against a conventional single-fidelity method. Under the same evaluation budget, MFBO consistently achieves higher similarity scores and more reliably recovers the prescribed responses. These findings demonstrate the potential of MFBO for data-efficient inverse design of stochastic architected materials, particularly where high-quality data are limited but lower-cost surrogate models are available.