Laser-driven proton radiography provides electromagnetic field mapping with high spatiotemporal resolution, and has been applied to many laser-driven High Energy Density Physics (HEDP) experiments. Our report addresses key questions about the feasibility of ion radiography at the Z-Accelerator (%E2%80%9CZ%E2%80%9D), concerning laser configuration, hardware, and radiation background. Charged particle tracking revealed that radiography at Z requires GeV scale protons, which is out of reach for existing and near-future laser systems. However, it might be possible to perform proton deflectometry to detect magnetic flux compression in the fringe field region of a magnetized liner inertial fusion experiment. Experiments with the Z-Petawatt laser to enhance proton yield and energy showed an unexpected scaling with target thickness. Full-scale, 3D radiation-hydrodynamics simulations, coupled to fully explicit and kinetic 2D particle-in-cell simulations running for over 10 ps, explain the scaling by a complex interplay of laser prepulse, preplasma, and ps-scale temporal rising edge of the laser.