The time of sunset, if that is what is causing the dits to lengthen and eventually drop off the air, should draw a line of possible locations. Actually watching the change each day, and ploting each line, should eventually result in a single point that should be fairly close to the location of the beacon. Of course local shading and altitude will cause the point to shift so that it is not necessarily exactly the location, but I bet it could be plotted to within a few 10's of miles.
Late edit. I also thought of something else. As the grey/dark line of sundown slants from SW to NE at this time of the year the line of sunup slants SE to NW also. Plotting both of those on a map, based on the stop/start times of the beacon, should yield a plot of the approximate location.