A brass rod which is a very close fit in the bore, pushed in from the muzzle end, should be able to catch the edge of the cartridge mouth. Wood dowel will work but could splinter, but that is less of a problem I'd wager with a ruptured .22 than say a lacquered steel milsurp round.
Wad up a cloth patch to go as a plug in the barrel first, that'll probably force it out, but you'll need to hammer the rod down to the chamber carefully. If you don't have a thick enough rod, wrapping patches to make it a good fit might work.
From the chamber end, push in a bronze brush that is a bit too large (.243?), only about half-way past the cartridge mouth. Then pull it back, hoping to catch the edge with the bristles which are too big and stiff (that's what *she* said!) to reverse direction in the bore.