Do you have a CO2 fire extinguisher nearing its hydro date? Use it to blast the chamber, just a single blast, then quickly use the extractor.
If not, get some dry ice. Wearing heavy gloves, chip off a small piece that will fit into the broken shell. Let is chill for a few minutes, then remove the dry ice pellet and use the extractor.
Or, do you have a CO2 inflator? The kind used for bicycles? That might work, discharge with the neck pointing down, it has about 5 grams of liquid CO2 in it. Prechilling the cartridge in a freezer also helps.
These methods will not harm the chamber. Remember, that extractor needs the case mouth to be a bit shorter than the chamber. If the sharp lip somes out with tiny shavings of brass, it might not work. Then a chamber brush is about the only way. Using a rapid chill method will still work with the brush. Chill the brush in dry ice pellets, this also helps.