D,
Apart from moving balance point up, the head will feel "lighter" while at the same time overal static weight will be heavier and depending on the grip, you will feel that. Eg, if you;re moving from a standard pistol to say a N66, that is a difference of 40 over grams I think. You will feel it unless you're like Ian (recall he can;t feel the difference in golf balls).
Another trick is of course shortening it which will remove swing weight. That of course changes playing lie angle which may or may not work for you. Not sure if your fitter has putter lie adjustment equipment.
IIRC, it takes 4 grams of weight at the butt end to remove 1SW pt. May be wrong so you'll need to ask google to verify.