Discrete complexes are a key tool to design stable and physics-compliant numerical schemes for certain classes of partial differential equations. A discrete version of the De Rham complex should reproduce the main algebraic and analytical properties of the continuous one. The Bernstein--Gelfand--Gelfand (BGG) method is a tool to combine copies of the De Rham complex in order to design other complexes. We build on our previous paper [1], where a BGG diagram was designed on polygonal meshes based on the Discrete De--Rham (DDR) framework. Polytopal methods are a class of numerical methods supporting meshes with polytopal elements of general shape. They are therefore less restrictive than classical numerical methods and allow one to naturally work on complex domains. The aim of this work is to show that the discrete BGG diagram naturally extends to physically relevant applications with minimal additional effort. In particular, we consider plate models, namely the Reissner--Mindlin and Kirchhoff--Love problems. To this end, we rely on previous work that makes explicit the relation between some BGG complexes and partial differential equations through the Hodge–Laplacian operator [2].