Ray chaos, manifested by the exponential divergence of trajectories originating from an originally thin ray bundle, can occur even in linear electromagnetic propagation environments, due to the inherent nonlinearity of ray-tracing (eikonal) maps. In this paper, extending our previous study of a two-dimensional planar ray-chaotic prototype scenario, we consider a cylindrical scatterer made of a perfectly electric-conducting azimuthally corrugated boundary coated by a radially inhomogeneous (ray-trapping) dielectric layer. For this configuration, we carry out a comprehensive parametric study of the ray-dynamical and full-wave scattering (monostatic and bistatic radar-cross-section) signatures, with emphasis on possible implications for high-frequency wave asymptotics (``ray-chaotic footprints’’).