Theories of in situ scanning tunnelling microscopy (STM) of molecules with redox levels near the substrate and tip Fermi levels point to 'spectroscopic' current-overpotential features. Prominent features require a narrow 'probing tip', i.e. a small bias voltage, eV(bias), compared with the molecular and solvent reorganisation Gibbs energy, E-r. However, a large V-bias is frequently needed for stable imaging. This applies particularly to in situ STM of redox metalloproteins, emerging as a new approach to imaging of biological processes directly in aqueous medium. We provide first an extension of previous theoretical work on in situ STM of redox molecules, to large bias voltages, \eV(bias)\ > E-r. Large bias voltages give tunnelling contrasts independent of the overpotential over a broad range, as both the oxidised and reduced redox levels are located within the 'energy tip' between the substrate and tip Fermi levels. STM here involves coherent two-step interfacial electron transfer between the redox level and the enclosing substrate and tip. We have also extended previous experimental in situ STM studies of the blue copper protein Pseudomonas aeruginosa azurin, adsorbed on Au(111), to cover a broad tunnelling current-overpotential range at a constant (large) bias voltage of +0.2 V. The current is found to be constant over a 0.25 V overpotential range, which covers roughly the range where the oxidised and reduced redox levels are located within the energy tip. STM contrast and apparent adsorbate coverage decrease outside this range. This could reflect in part redox processes of azurin, but also tip-induced disassembly of the azurin monolayer.