The evolution of cooperation is thought to be promoted by pleiotropy, whereby cooperative traits are coregulated with traits that are important for personal fitness. However, this hypothesis faces a key challenge: what happens if mutation targets a cooperative trait specifically rather than the pleiotropic regulator? Here, we explore this question with the bacterium Pseudomonas aeruginosa, which cooperatively digests complex proteins using elastase. We empirically measure and theoretically model the fate of two mutants--one missing the whole regulatory circuit behind elastase production and the other with only the elastase gene mutated--relative to the wild-type (WT). We first show that, when elastase is needed, neither of the mutants can grow if the WT is absent. And, consistent with previous findings, we show that regulatory gene mutants can grow faster than the WT when there are no pleiotropic costs. However, we find that mutants only lacking elastase production do not outcompete the WT, because the individual cooperative trait has a low cost. We argue that the intrinsic architecture of molecular networks makes pleiotropy an effective way to stabilize cooperative evolution. Although individual cooperative traits experience loss-of-function mutations, these mutations may result in weak benefits, and need not undermine the protection from pleiotropy.