I've actually done something similar with one of my chess sets because many of the pieces in the set had loose, rattling weights. Of course, I never hollowed out the pieces myself and the lead weights inside were ones that came with the pieces themselves, not fish weights. Also, the pieces were made of plastic, not wood (not sure if this makes a difference).
I first removed the felt, removed the weight inside, filled the inside with a bit of RTV silicone, placed the weight back in, placed a bit more silicone into the hole to level the base of the piece, then waited for the silicone to dry. I replaced the felt using one of those felt stickers usually used for heavy furniture. I've found that these stickers don't stick very well to silicone, so you'll have to make sure to peel the silicone off the base of the piece after drying.
I'm not really sure if the silicone will work with wood. Seeing as your wooden set is on the expensive side, I'd wait until more comments before committing to anything.
I received this set for Father's Day http://www.chess.com/eq/chess+sets/maple--walnut-chess-set-with-storage.
The only thing I don't like is the weight of the pieces. Does anyone know how/where to purchase weights? I was actually thinking of getting a package of fishing weights. I could drill a hole, glue the weight in and replace the felt. Has anyone ever done this?