An image (patterns, screens, etc) outside the web width will do wonders in minimizing the accumulation of ink on the roller ends. Just how much image to use for best results seems to vary, I have seen anything from a light screen to solids being used, I do not know how the correct density of image area was determined.
I think the mourning bars are a good idea.
How good is the zero point when the ink fountain keys are closed in the extreme end positions. If you keep feeding ink where it is not wanted it will eventually build up and become a problem. You could see if you maintenance staff can fit an air bar over the inking system. You could use air to get rid of excess water in the non image areas.