Lead is a high-Z material, which provides a lot of electrons to interact with photons (and charged particles). It’s not good for stopping the neutrons that pose the danger here, because of the disparity in mass between the shield nuclei and the neutrons. For storing fissile materials in proximity, the optimal material is something with a high absorption cross section, like boron. Water and plastic are good at removing energy from neutrons and preventing leakage in typical large shielding scenarios, but in this case they would increase the danger, because they thermalize neutrons efficiently (as does a workers body) and this increases fission interaction probabilities and reduces the margin to criticality. For spent nuclear fuel, something like a boron-carbide plate is used. LANL may use something similar.