Maybe just recessing the contrast control a little further would be the best solution here.

Going to a software-controlled contrast control is:
a) more expensive
b) much coarser control (i.e. it would be stepped, and as I'm sure you've found, the ideal setting is always between two steps :-)

A software-based contrast control is only usable when there is already sufficient contrast for the display to be visible. Consequently there needs to be some minimum contrast guarenteed by the hardware, probably as an internal factory adjustment. The software-based control ends up just being a 'fine-tune' for contrast.