Simulations of the passage of eukaryotic cells through a constricted channel aid in studying the properties of cancer cells and their transport in the bloodstream. Compound capsules, which explicitly model the outer cell membrane and nuclear lamina, have the potential to improve computational model fidelity. However, general simulations of compound capsules transiting a constricted microchannel have not been conducted and the influence of the compound capsule model on computational performance is not well known. In this study, we extend a parallel hemodynamics application to simulate the fluid-structure interaction between compound capsules and fluid. With this framework, we compare the deformation of simple and compound capsules in constricted microchannels, and explore how deformation depends on the capillary number and on the volume fraction of the inner membrane. The computational framework's parallel performance in this setting is evaluated and future development lessons are discussed.