I was fitted for a putter several years ago, it was a really cool process. During the fitting we tried weighting the grip. It seemed to have "helped". I played with that putter for several years. In hindsight I don't think that it made a significant difference.
My car was broken into and the set of clubs that included that putter was stolen. When I replaced them I upgraded to a Scotty Cameron. I did not have the grip weighted.