What is the best way to permanently bond a driver tip to a stuck screw head?

Question

I have been trying to close the two screwdriver-operated stop-check valves (visible on the left and on the right in the picture) of a Moen single-handle shower to cut the flow of water and allow me to replace the shower cartridge (as I live in a condo, this would simplify my life as there isn't a water shut-off for my apartment).

These valves have not been operated for decades, and while I was able to get the cold water valve working with my impact driver, the hot water side did not budge by a single degree. Several attempts with screwdrivers, impact drivers and screw-biting pliers only resulted in a screw head that is stripped both in its slot and around its circumference.

As a last attempt, I am thinking of permanently "embedding" an impact driver tip into the screw head, by way of some glue or cement (I have no welding equipment). I was wondering if anyone has ever tried something similar and if there is a cementing material with the required torsional strength that, if applied just on the small surface of the screw head, could withstand the high torque necessary to unstick the valve. In the past, in similar situations where high force was applied on a small area (like a luggage zipper) I have used JB Weld SteelStik Steel reinforced epoxy with great results. I am just not sure whether it is the best choice to bond with the screw material, which I believe is brass.
Any suggestion is highly appreciated.
Below is a rather pathetic attempt at illustrating pictorially what I would like to do.

EDIT: Here is a pic that shows how badly the screw has been eaten up by my attempts. I added two blobs of epoxy to try to reconstruct the screw slot walls. I'll see if it does anything good after it cures in several hours. Note: the epoxy was useless, chiefly because of the penetrating oil that had been sprayed earlier on the screw which compromised any possibility of adhesion to the screw head.

Answer 1

The torque required to turn the screw will be too great for any of the locally purchased epoxies. Take a rotary tool, like a Dremel, with a cutoff wheel or a hacksaw and deepen, widen the slot on that screw so a larger screwdriver will fit in there. Try loosening that nut behind the screw just a bit to break and buildup of lime.

Answer 2

These valves have not been operated for decades, and while I was able to get the cold water valve working with my impact driver, the hot water side did not budge by a single degree.

Emphasis mine.

Go to your water heater. Turn off the valve exiting the water heater. Run the tap on the hot side, ideally from a faucet lower than your shower valve, to alleviate the pressure. You should then have a dry hot water side and you can disassemble or replace the hot side shutoff as necessary for your shower cartridge.

Answer 3

This is not a novel answer to the question I posed, but just a little summary of what was achieved by applying the suggestions received, in the hope it might be useful for someone who ends up in a similar situation. First of all, thanks to everyone who provided useful contributions and a particular shoutout to @FreeMan whose recommendation of replacing the useless Gunk Liquid Wrench with PB Blaster was the key step in getting the stuck screw free. It was, however, a bittersweet win, because, the hot water valve, unlike the cold one, did not manage to completely arrest the water flow and hence did not allow the cartridge replacement. This might be due to some residual corrosion buildup that still prevents the full travel of the screw. However, in my opinion, it is more likely due to a deterioration of the rubber around the valve puck caused by a combination of age and temperature. I base this opinion on an inspection of other two showers in the same apartment complex: both of them had a cold water valve that could be made operational relatively easily with an impact driver and a hot water valve that did not budge by a single degree.

The best procedure to unstick the screws is as follows: spray some PB Blaster on the valve assembly around the screw (as sparingly as possible, because the more you spray the more you will see coming out when you open the spout) and spend a couple of minutes tapping around the screw head with screwdriver and hammer from different angles, as it might dislodge some of the gunk inside. To minimize the damage to these rather delicate brass screws, select very carefully an impact bit that fits the screw slot as tightly as possible and operate it with an impact driver with light pressure on the trigger in such a way to apply mostly just the stall torque of the driver and release impact hits only in a slow controlled way until the first movement of the screw. After the first movement is achieved, use a hand screwdriver (or even better one of the fantastic Engineer pliers especially designed for screw removal, preferably the locking type) to achieve full screw travel. To verify whether or not full travel has been reached, pull the shower handle and turn it in the appropriate direction to check whether there is any flow of water from that valve.

Some final details on those stop-check valves: during operations they are kept naturally open by the line water pressure and the screw closes them by pushing a spring-connected rubberized puck to block the water inlet. If the rubber deteriorates, the puck might get stuck somewhere along its travel. For anyone having the same setup it might be useful to know that the part ID is Moen 12318 Stop-Check Valve Kit and read this review, which provides a great explanation of the particular theory of operations of these stop-check valves.