Follow Erik’s suggestion for the nonlinear Static Structural part of your analysis to get the behavior of the cables going slack under certain loads.
You observed that Large Deflection is not available in Modal. That is because it is a linear analysis. When Workbench has a nonlinear model, it must linearize it before submitting the model to the solver. Nonlinear contacts that can change from an open state to a closed state during loading of a nonlinear Static Structural model must be converted to be either open or closed during the Modal analysis and Ansys has rules for deciding on how to convert each contact. Similarly, if there is initial tension on the nonlinear cable, I expect they will behave like springs in Modal and any subsequent linear analysis such as Harmonic Response. So even if the deformation in the linear Harmonic Response would have made the cable go slack in the nonlinear Static Structural, I expect that cable will generate compressive forces.
To get a more accurate dynamic response to something like earthquake ground motion where the deformation would cause the cables to go back and forth from tensioned to slack during the simulation, you will need to do that in a full Transient Structural and not use the linearized MSUP methods that use a Modal analysis.
However full Transient Structural analysis takes longer to solve and longer to postprocess. You could start with two linear models, one with cables tensioned and one without any cables to get an upper and lower limit on the linear response of the structure before starting the nonlinear full Transient Structural analysis.