Try match grade ammo. Most of my cutsom rifles do exactly the same thing if you try to feed them factory ammo. You are probably going to have to turn case necks and use bushing dies. Might also have to bump the case neck back just a smudge.
I am not trying to be disagreeable, but I seriously doubt the chamber is "out of round" on a SPS. I am not a Remington fan or a Remington hater, but the SPS is jam up.