If the bullets came apart in the gun they had to be gripped by the throat in the barrel. You could try to rotate or remove some bullets from the cases in the box of ammo with your fingers to get an idea of how much that would take. On my gun it leaves a ring of scuff marks around the bullet just outside of the case mouth. You may need a magnifying glass to get a really good look.

I measured a half dozen Sierra match bullets and they were right on at .3080 down to .3078. I didn't see any over .308.

I see the scuff ring on rounds I hand load when I am trying to figure out the seating depth for bullets. With the tipped Match King bullets the rifling doesn't engage but the throat does. When I chamber factory FGMM, I get a scuff about half way around the bullet right outside the case mouth. Since it is half way around that must mean that the chamber and throat are slightly non-concentric. Nothing is perfect so the FGMM is also occasionally non-concentric and also probably varies in bullet seating depth too. Very small amounts here, but sometimes a round is more difficult to chamber.

I did measure another brand of match bullet (Nosler) and they were about .3075"-.3077", consistently smaller than the Sierra pieces. It was enough to notice the difference in feel pretty easily when seating them.

You could try some other brands of ammo, maybe more hunting oriented and see if you find something with a slightly smaller size. Then you can send your gun in to Savage over the winter ( do you have winter? ) with more info about ammo that works and doesn't etc. They would fit a new barrel.