Where to start....

First, I can't say I've ever seen a Savage Axis with a rear tang before. Every one I've ever seen has the safety/trigger group as a sub-assembly mounted via a pivot pin like the one pictured below. If yours is different please post a pic.

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Second, the placement of the rear action screw is not an issue and hasn't been for the hundreds of thousands of Savage 110's and Axis rifles made over the last 60 years. I would actually argue Savage's placement is better as it offers more bedding surface area at the screw location than if it were located on the rear tang like others, but that's another debate for another time. The Axis' design makes bedding a little more difficult, but it's doable.

Third, I would be willing to be the shift in POA you are seeing is due to the flex in the wrist of the stock rather than a bedding issue. It's a known weak point in the stock and myself and others have proven multiple times over the years that even slight variations in grip pressure can affect POA and POI.

As others have noted above, trying to "fix" the factory stock is all but a waste of time. The inherent flex in the stock makes bedding it a fruitless effort. Filling in the forearm with aluminum rods and bedding compound to try and stiffen it offers little gain as the rod would have to pass through the recoil lug pocket to actually stiffen it as that's the weakest flex point. Even after doing that there's still the huge amount of flex in the wrist area that there's no way to eliminate. So yeah, it's purely a lost cause. It's plenty suitable for a basic deer rifle, but like most turds no matter how much you try to polish it up it's still going to be nothing more than a turd.

So, unless you reconsider the option of replacing the stock you've pretty much tied your own hands here.