Flexible slender structures, like cables, exhibit complex constitutive mechanical behaviour. To enable more realistic simulations, this work proposes an enhanced constitutive model for three-dimensional geometrically exact Cosserat rods under bending loads. This constitutive model takes into account nonlinear and inelastic effects. The nonlinearity is modelled by means of a characteristic function that makes use of curvature-dependent bending stiffness values. The inelasticity is described by Prandtl-Ishlinskii hysteresis operators. For a set of boundary conditions, an algorithmic approach is used to compute quasistatic equilibrium configurations by iteratively updating the constitutive parameters defining the characteristic function and the hysteresis operator. To identify the constitutive parameters specific of a cable, an inverse problem is formulated based on experimental data collected from a bending test. In our case, a test rig is set up to perform approximated pure bending tests. Based on the measured bending moment, a data fitting problem is solved for a sequence of equilibrium states, and the simulated bending moment is evaluated through the proposed constitutive law. The comparison of simulated and measured bending moments will be shown for various study cases, such as high-voltage cables and FLY cables. The results are promising, and we aim to validate this enhanced constitutive model with more diverse and complex bending loading scenarios.