Systematic three-dimensional finite element (FE) simulations are carried out to study the ballistic protection performance of double-layer sandwich plates having metallic pyramidal lattice truss cores filled with ceramic prism insertions and void-filling epoxy resin. Both normal and oblique projectile impacts are considered in the FE simulations that are validated against experimental measurements. The ballistic limit velocity, the energy absorbed by key constituting elements and the critical oblique angle corresponding to the transition from ballistic perforation to projectile embedment are calculated. As the oblique angle is increased, the evolution of deformation and failure in the double-layer plates as well as the underlying mechanisms are explored. It is demonstrated that the proposed double-layer sandwich plates outperform both the single-layer sandwich plates and the homogeneous (monolithic) metallic plates having equal total mass, and the top layer (the ceramic insertions in particular) of the double-layer configuration plays a more dominant role in energy absorption.