Angel hair? Guessing you mean some sort of loose glass fiber.
Anyway, looked while trying to avoid some stupid work this morning: At least some Choate stocks are glass-fiber reinforced polypropylene. PP is very hard to glue to. It's a "slippery" plastic, that not just water but other solvents, and glues, run off of instead. Most cyanoacrylates (super glue) won't even work, and require special primers. This is a material property; degreasing or even sanding won't change it.
Reportedly, 3M Scotch-Weld DP8005 will work. Never tried it. I'd consider experimenting before cutting. Get some glues and try to attach the head of a screw to the stock somewhere innocuous, like under the buttpad. Read and give full cure time, then try to break the screw off. You'll be able to judge if the force required was enough for your project, but it may be impossible to get the bond you want.