If you bleach the stock, you will need to stain it. The water will also swell the grain, so if any inletting is close around the receiver, you may find it no longer fits after a soak. The grain will definitely be raised. You didn't say whether the wood was good and whether it was worth preserving the grain and color? Also, if moisture in the wood is changing its size enough to shed the finish, I would worry about it splitting in water. This depends on how absorbent it is? With the military stocks, they usually aren't tight fitting and raising the grain helps to get birch and some other woods to take stain and finish better. Also replacements are usually easy to get hold of.
I use a different method: I take the stock outside and use a conventional paste stripper. When the finish loosens, I use those plastic paddles sold for applying body putty or fiberglass as scrapers to get the bulk of the old finish off. Steel wool gets much of the rest, and I use a natural bristle brush in a wood handle to work out any softened finish stuck in the checkering. I usually make at least one additional application of the paste. To remove the last of the paste, I switch to a liquid remover and dip steel wool and the brush in it to wash the residue away completely. Wear disposable Nitrile gloves and eye protection and do this outdoors.
Let the stock dry (I wipe it off outside, let it sit an hour, then hang it in the garage for a day). Next I put the stock in two or three layers of heavy trash bags together with a large container of fresh desiccant and let it dry out for a month.
To finish, start in the morning of a pleasantly warm dry day. I do this over newspaper in the garage. Sand everything with 240 grit paper, then apply Deft Danish Oil using 320 grit wet/dry paper as the applicator. Get a colored version if you need stain. Keep the surface wet half an hour by reapplying it with a brush to any dry spots that appear.
After the half hour, wipe it down well with a rag. An hour later, wipe down again, then apply fresh Danish oil, this time using 400 grit wet/dry paper. Same routine. Keep wet half an hour, wipe it, dry one hour. The third application is made using 600 grit wet/dry paper. Same routine. This time after wiping, check every hour for the rest of the day for the reappearance of wet spots and wipe dry again. Check again the next day, and rub it well with a rag.
It should go without saying that checkering cannot withstand all the sanding. Just use a brush. The sandpaper application makes a smooth satin finish, but checkering masks that anyway. If you need to, you can fold the wet/dry in half, abrasive side out, and use the edge this creates in the bottom of the checkering grooves, but beware it is easy to go overboard and hurt the checkering.
Deft and other companies make polyurethane final finishes to go over Danish oil if want gloss. None will penetrate and make the wood as weather resistant as this approach does. The only better method I am aware of was the one used by the AMU on M14 stocks for match rifles. That involved baking the stocks dry in an oven (the desiccant is kinder and gentler) the puting it in a container filled with epoxy finish and pressurizing it to 100 psi. Lot of bother and expense for the amateur.
Nick