I thought about Butlers R suggestion of using a tap. A 5/16 tap would be too small probably, an 8mm tap would be slightly larger. It might work. Trying to start the tap without scarring the chamber is scary especially with out a guide. I have access to a lot of machine tools here. If I was to try this, I would shorten the cartridge that this broke off from and drill a concentric hole (in a lathe) through the primer end just big enough to clear the tap body. This would get the tap started straight. To turn the tap, I would mill a 6mm slot in the end of a 5/8 bar, since getting a tap wrench that long and skinny would be difficult.
To drive stuff out of the chamber I prefer dropping a piece of 1/4 bar stock from the muzzle end. Sometimes it takes several drops but it seem to me less likely to scratch the bore.
I may be more of a chicken than Butlers R,  if I did not have the tools to do as described I would take it to a gunsmith that has a lathe and mill.