The de Haas-van Alphen effect describes the periodic oscillation of the magnetization in a material as a function of an inverse applied magnetic field. It forms the basis of a well established procedure for measuring Fermi surface properties, and its observation is typically taken as a direct signature of a system being metallic. However, certain insulators can show similar oscillations of the magnetization from quantization of the energies of electron states in filled bands. Recently, the theory of such an anomalous dHvAE (AdHvAE) was worked out, but there has not yet been a clear experimental observation. Here, we show that the inverted narrow gap regime of $InAs\; /\; GaSb$ quantum wells is an ideal platform for the observation of the AdHvAE. From our microscopic calculations, we make quantitative predictions for the relevant magnetic field and temperature regimes, and we describe unambiguous experimental signatures.