From your photos It appears to me that previous owners have carried out some unsympathetic sharpening. In other words – as you’ve worked out – the blade and the sole-profile don’t match and the mis-match allows the bed profile to bottom out and stop cutting.
It is easy to attack the blade in isolation and to try reforming the profile, but that plane is definitely pre-war in age…… probably pre-Boer-War….. and personally I would start with the sole. Just reforming the blade profile in isolation is unlikely to give good results. Give it a complete re-hab!
Ensure that the whole of the base profile is dead flat from toe to heel. Lay a straight edge along each part of the curve and the side sides and sight it for gaps. Given the age of the plane I would be surprised if it is flat. Carefully running a straight edge along it with some carbon paper attached will help identify any high spots and it is wise to deal with these first. Nothing more than a gentle scrape with a squared blade. NEVER use sand paper! It is vital that the area of the sole that surrounds the moth, back and front, is in contact with the work when you have finished. If this part is raised off the work the iron cannot cut correctly.
When the sole is clean and flat end-to-end, reprofile the iron. This means undoing previous sharpening work and filing the edge back so that it exactly matches the sole profile when you sight along the plane. This will inevitably result in flat sections along the cutting edge and it is now that you re-file back the bevel, ensuring that at each point it is at the same angle to the profile. Polish the back and finish it with fine slip stones to get a good cutting edge.
A lot of work, but it will be worth it to give that old tool a working life back. Good luck.