You might also try removing some weight from the stock by drilling out some wood. That will shift some of the balance forward a bit. I don't think that it will help much because wood isn't as heavy as you might think and stock weight doesn't have the same mechanical leverage as barrel weight. Normally, fiddling with stock weight is very slight fine tuning of a gun that is almost right and just needs a tiny touch. It doesn't sound as though your gun are there.
Long stocks are a standard procedure on light weight subgauge guns.
Though I hesitate to recommend the easy way out, this might be one of those times. Consider selling the gun. Rich Cole at Cole Gunsmithing, Rt. Try Rich first. I believe I saw Joel Etchen handling some at a good price.