A rifle much over 10 lb. becomes a challenge to field portability, in my view, but if he insists, you can put weight into both the forestock and butt stock until it balances where he wants it. In most modern guns that point usually falls between half to one and a half hand widths in front of the trigger guard with a full magazine and a closed action. Some people prefer a muzzle-heavy gun, but that isn’t as common as it once was.
To make a butt stock counterweight, start by drilling a hole in the buttstock large enough for a length of pipe. Mix a little hydraulic cement to plug the last 1/2" of the pipe, then rest it in a tin on a brick filled with wet sand deep enough to keep the pipe upright and pour in enough molten lead to make the balance weight. After it cools, glue the pipe into the hole in the buttstock. If you don't want it permanent, pack sponge foam in the bottom of the hole and at the back of the weight so it will be retained by the butt pad plate and won't shift or rattle.
I second Countryboy's susggestion for stiffening rods in the front. They are a really good idea with a bi-pod. The thermoplastic factory stock on my Savage 10FP, when resting the rifle on a bi-pod, flexed the forestock enough to rub the bottom of the barrel and spoil accuracy. It is a common problem. Most people epoxy bed in a synthetic stiffener (fiberglass or carbon composite shaft from damaged arrows, for example) to correct this because they are avoiding extra weight, but in your case the steel weight is desirable. When you glass the steel in, you can mix #9 shot in with the glass bedding compound to increase its weight, too, or you can use one that already has a lot of steel in it, like Brownells’ Steel Bed.
Personally, I like the recoil pad and muzzle brake suggestions as first go-to over increasing weight beyond 10 lbs. Your customer could also try wearing a
PAST Sorbathane recoil pad on his shoulder for added buffering. I am surprised that at 10 lbs. the rifle’s recoil still bothers him, but some people are more sensitive to it than others. The muzzle brake-type recoil reducers work best with large powder capacity cartridges, so that approach should do well in this instance.
One thought would be to double-check the stock's fit. Some stock shapes jab some people in the cheekbone rather than recoiling away. Make sure that isn't what's getting to him rather than straight shoulder recoil? If so, the stock is the wrong shape for this fellow. Straight shoulder recoil should be addressed by the pads successfully.
Nick