If the rifle shoots well, think about reloading for it.
There could be other issues besides the chamber. Like the locking lugs setting back the lock up faces on the receiver. This happened once in a while to re heat treated receivers or late war receivers that weren't heat treated properly.
When many of these old war horses were converted to fine sporters, some of them were heat treated to engrave. I had a lovely sporter that was done in France after WWII.
It had double set triggers, incredible engraving over most of the visible sections of the receiver and half way down the barrel. It was accurate and pleasant to carry but a bear to shoot.
It had a very bad problem, exactly what the OP described. The primers would back out about .025 with factory loads and the shoulders would round out.
I really liked the rifle, so I hand loaded for it and increased the chamber pressure to around 45,000psi. Yeah I know not reccomended.
The shoulders moved ahead about .025 on the fire formed cases and I backed off the powder charge to a slower burning powder and lower pressure. Never had a problem with the rifle or the primers backing out after that.
That rifle, will never see another owner. It will be parted out and the receiver will be destroyed. I don't shoot it any more as it is light and the recoil is brutal as mentioned above.
The triggers will look good on a 7x57 build I have in mind on a Brazilian Mauser that was previously altered.
This may or may not be the case with your rifle. Take it to someone that can see inside to the locking faces in your receiver. Sometimes a dental mirror can show up the indentation but not always.
If you're only going to shoot North American factory ammunition out of it and the rifle shoots well, don't worry about it. It won't hurt anything with those loads.
If you decide to hand load, or use Norma factory ammunition, take it to a qualified gunsmith to have a close look at.