We investigate the nonlinear viscoelastic behavior of a biomimetic scale-covered beam in which shear-dependent complex fluids are trapped between overlapping scales under bending loads. These fluids mimic biological mucus and slime layers commonly enveloping the skins found in nature. An energy-based analytical model is developed to quantify the interplay between substrate elasticity, scale geometry, and fluid rheology at multiple length scales. Constant strain rate and oscillatory bending are examined for Newtonian, shear-thinning, and shear-thickening fluids. The analysis reveals unique, geometry- and rate-dependent viscoelastic response, distinct from classical mechanisms such as material dissipation, frictional resistance, or air drag. Energy dissipation is shown to emerge from a nonlinear coupling of tribological parameters, fluid rheology, and system kinematics, exhibiting distinct regime-differentiated characteristics. The model captures the competitions and cooperations between elastic and geometrical parameters to influence the viscoelastic behavior and lead to geometry and rheology scaling laws for relative energy dissipation. The pronounced nonlinearity in the moment-curvature relationships, along with the geometry-controlled regimes of performance, highlights the potential for using tailored and engineered complex inks for soft robotics and smart damping systems.