I think most glue will leak through the felt. Try an adhesive spray onto the felt and then place it on the bottom. I'm just guessing but my thoughts are that the spray isn't as thick as traditional glue and it might not leak through. Do a test on a piece of felt.
As for the weights in the future, you could always try a fishing store for weights. They may even mold some into a size that you may need
This topic pops up from time to time with various ideas and solutions. I have a favorite set of Drueke #35 pieces that my grandkids used as toys one day when I was away and the adults were not watching them. I lost one weight in a knight, and about half of the felt bases were torn loose or removed.
I have looked in vain for weights that might fit inside the knight. I haven't found any commercial source of variably sized weights or slugs and have searched in vain for a solution online.
I finally found that U.S. quarters were just a bit smaller than the base of the knight and I taped a stack of 4 of them together which resulted in a fairly snug fit in the base and within 1 gram of the same weight as the other knights. The stack was glued together with Elmer's glue, taped and then glued into the base to prevent rattling.
That problem solved, I turned to the bottom felt. I found that synthetic felt is very cheap, durable and the right color for replacement on the bottom, but I am searching for ideas on glue to attach the felt to the bottom which consists of both plastic and metal contact surfaces. Elmer's works OK except that it is too thin and soaks through the felt. When it dries, it creates a hard surface which is little better than no felt at all.
So the question - what glue is suitable for attaching the felt (essentially plastic felt) to both plastic and metal without running through the felt and hardening on the bottom? I'm afraid a heated glue gun would melt either the felt and/or plastic. Elmer's is too thin. Can someone recommend a suitable bonding agent that will stick to these surfaces without compromising the soft felt bottom?