I recently added "Great Stuff" foam to my Tikka T3 Lite, not to add weight, but to reduce noise from scraping against brush in the field and to insulate the stock to help make it warmer to the touch in cold weather. It worked fine.
I didn't want to add weight, so didn't think of adding lead shot, but if it were to be added, a concentration located parallel to the bore might help to keep recoil straight back.
People are like rifles. Some are tried and true, having great eyes, personality, and fun to be with. Others never seem to hit the mark with you. Still others go off half-cocked. Still, it's nice to know most of them.