In my experience, the problem may well not be fixable by tweaking the rear triangle. The dropouts are probably at different heights. It doesn't take much error at the dropouts to make 2mm at the rim, less than half a millimeter. The string test is okay, but you have to recognize that a lot of things could be out of alignment and you have to average out the faults to see what's really going on. It's the Heisenberg uncertainty principle of frame alignment.
The engineer in me has come to realize that frames are going to be out, especially 40 year old Italian frames. I have an alignment table and a number of frames that I would check on it if I had the energy. Mostly to see if I can figure out issues with the handling though.