Folded protein-based hydrogels are a novel class of biomaterials which combine the useful viscoelastic properties of individual proteins together with the prospect of rational design principles. Although the macroscopic properties of these materials have been well studied, there is a paucity of understanding of their mesoscopic formation mechanisms, especially given the differences in building blocks compared to biopolymer hydrogels. We present the results of a simulation study into the growth of polymeric networks of chemically cross-linked folded proteins that form the structural backbone of these hydrogels, observing how experimentally controllable parameters affect the resultant network growth and structural characteristics. We show that the initial volume fraction emerges as a dominant parameter at the network level but that the properties of the single protein remain important. We ultimately show that we can tune the properties of a monodisperse protein hydrogel network only within limits which are dictated primarily by implicit diffusion time scales.