Air-jet weaving involves complex interactions between high-speed air flow and slender textile yarns, often consisting of tens or hundreds of individual fibers. The characteristic length scales in this process span several orders of magnitude: from micrometers at the fiber level to meters at the machine level. Resolving these fiber-level details in machine-scale simulations remains therefore computationally infeasible. Yet precisely these fiber-scale characteristics determine the yarn's response in high-speed air flow, influencing both the aerodynamic drag forces, affected by yarn hairiness, and mechanical response, such as non-linear stress-strain relations, originating from the internal microstructure. Recent work [1] introduced a multiscale methodology that incorporates fiber-scale effects into macroscale Fluid-Structure Interaction (FSI) simulations. On the macroscale, an Actuator Line Method (ALM) [2] for the flow is coupled to a beam-element structural model, without explicitly resolving the individual fibers. Instead, the effects of the yarn's microstructure are incorporated in aerodynamic force coefficients, obtained from microscale Computational Fluid Dynamics (CFD) simulations on µCT-based yarn geometries, and the mechanical material model, derived from microscale structural simulations. This approach allowed for the first successful high-fidelity simulation of the launch of a hairy yarn from the main nozzle of an air-jet weaving machine, which is not possible using traditional methods. For a smooth nylon monofilament yarn, a reduction of 60% in computational cost was observed, while still predicting a similar yarn velocity. Despite these improvements, the high-fidelity model remains computationally demanding, requiring several days to weeks of runtime. To address this limitation, an acceleration strategy is proposed based on asynchronous, or multirate, time stepping. In a first test, the CFD solver operates at a coarser time step (50 µs), while the aerodynamic forces on the yarn are still evaluated every structural time step (5 µs). As such, it exploits the slower evolution of the flow field, with inlet pressure oscillation periods of milliseconds, compared to the yarn dynamics. Preliminary results show an additional 75% reduction in computational time, substantially improving feasibility for optimization, parameter studies, and uncertainty quantification (UQ).