The need for a fast transition towards clean electricity generation pushed large investments in high-risk, high-impact technologies such as floating Airborne Wind Energy Systems (AWESs), which are expected to be highly cost efficient with respect to state-of-the-art offshore wind energy technology. Current research on the matter does not address the design challenge of a tailored floater, necessary to suit at most the unique features of such systems, and foster their industrial and commercial development. This study proposes a simulation-based design optimisation framework for the floating structure and associated mooring system tailored to offshore AWES deployment. By reviewing conventional floating wind energy solutions and analysing expected AWES loading conditions, a spar-like floater and taut mooring system are proposed as the baseline design for a numerical case study. A state-of-the-art first-order potential flow model is employed for the floater, coupled with a novel quasi-static model for the mooring system, to simulate system motions in the frequency domain. The design optimisation aims to minimise capital expenses and wave-induced motions, with constraints on the strength and fatigue lifetime of mooring ropes. A multi-objective genetic algorithm is used to explore the design space and approximate the Pareto front, with sensitivity analyses on the latter revealing the significant impact of mooring line materials on system performance. Verification of the optimisation results includes an assessment of aerodynamic loads using a decoupled AWES model and time-domain simulations in OrcaFlex for selected configurations under various wave conditions. Although the quasi-static mooring model showed coherence with dynamic simulations, optimal configurations deviated significantly from the initial spar-like design, indicating potential benefits from alternative baseline designs. The impact of AWES aerodynamic loads on deck motions highlights the need to integrate them into the design process. Despite these challenges, the proposed simulation-optimisation framework shows promise as a powerful tool for the preliminary design of floating AWESs, paving the way for future refinements.