Cascomite I have used in boat building and its good and strong. But you need tight joints. I would use as Damascus says good old fashioned 24hr Araldite, or something like West Systems or SP epoxies - these are a thin epoxy resin to which you add fillers to get appropriate consistency and strength. You can make it runnier than araldite so the epoxy soaks in well. Key with any repair is to make sure it's not oil contaminated. Middle of a hand of a stock should n't be too bad, but depends a) on original construction, b) how it was finished and c) whether it has had lots of gun oil applied and then stored butt down. Acetone is good for cleaning oil out of the wood first.
Do Brownings have their stock attached with a stock bolt? if so you might need either drill it out and put in a sleeve, or use it as the reinforcement, but thoroughly wax it first so it doesn't stick. Brass is good, but may be a bit heavy - no reason not to use say carbon fibre rods if you want light weight.
Must admit a lot of modern fancy stocks show little understanding of wood engineering. Mostly the grain is laid out incorrectly, and frankly mostly they are done for show, rather than strength. Look at stocks on old fine guns - they all have nice strong and straight grain. Burrs break - fine for veneers, but not for anything structural.
