Does that huge crack in the stock visible from the outside through to the bayonet groove seem to have anything to do with it or is it just sitting there pointing out that issue?
Is there any wood missing on the inside? It may be possible the movement actually caused it when firing...due to an ill shaped stock. Pics of that without the rifle might help. If anything obvious isn't seen...folks with micrometers could check and see if the stock was just bunk from the factory. I wonder if you could get a decent recording of firing it, you would hear a sound similar to swinging for the bleachers with a cracked bat.
Who knows, this might be the reason it was a bring back...if accuracy is affected by this, the previous owner might have lost a duel.
Something I would try, swap out the stocks and see if the problem translates over to the other gun. That will at least tell you whether it is the stock that is the problem, or the stock retainer. Granted there is quite a few years between the two, and variances in tolerances with the SKS are well known...it shouldn't be THAT much of a variance. Of course you can only do this with the bayonets extended, but those probably wouldn't have anything to do with it.
For a non damaging/permanent fix, you might try plumbers putty packed into the stock retainer. It won't alter the historical rifle, and should eliminate the movement....just don't use too much or it will squish out and be visible. It shouldn't discolor anything, and should harden a little over time. Heck, you could even try staining the putty before use to try match the stock color even if it does show a little... I don't know if staining the putty is possible...but you will have plenty left over. I have never seen the bottom of a tub of that stuff.
Oh...and welcome from Montana!