You could try a mixture of acetone and transmission fluid. Seal up the chamber end with a rubber stopper, pour the atf/acetone down the muzzle and let sit for a couple days. Stick the barreled action in an old paint can to contain leaks. The atf/acetone mix is usually 50-50 I save the larger pill bottles some of my meds come in so mix the atf/acetone 50-50 in that so you don't waste either. And yes a broken shell extractor for the 303 Britishicon cartridge is a good investment. If you do get the piece of brass out, shine a strong light in the chamber to check for damage. Some Maroon 3M sanding/polishing pads cut up and setup in a split dowel with some oil will help remove any burrs and scratches..And be sure to thoroughly clean the chamber and barrel before you test fire. Frank