This does NOT look like a shaft machined to ASME/SAE standards for tapered and threaded propellor shafts. For instance, I've never seen a single prop nut on a shaft without a cotter pin to secure it. Nor a washer between a prop nut and the prop. Sure, lots of double-nutted installations, but never one with only a single nut, without a cotter pin. And you call this washer a "lock washer". How so? Not standard operating procedure for securing a propellor to a shaft, for sure.
I suggest you investigate different double prop nut installations, and the correct assembly procedure, such as: https://www.passagemaker.com/channel...t-myth-busting
Should a new half-height nut properly torque down against the prop (properly seated on the taper, of course), then the assembly may work as designed. This might be a stretch, given the extent of damage to the shaft immediately behind the propeller. However, if the appropriate bronze half-height nut torques properly upon installation, all should be well.
Assuming, of course, that there are at least 1 diameter's worth of threads left to engage with the full-height nut! The thread length looks too short, at least to the naked eye. It is entirely possible that someone in the distant past miss-machined your shaft. Is the stbd shaft/prop/nut similar?
Drilling the existing shaft, and using a single castellated nut and cotter pin is a possibility. However, castellated propellor nuts are significantly thicker, and you may not have enough shafting under the "castle" to accept a drilled hole and cotter pin. Assuming, again, that the single nut will accept full installation torque.