I would try adding a bit of JB Weld in the hole. Just enough to seal it up. Don't add so much that it runs, otherwise you risk having a much bigger mess... Once that is hardened, you can use a bolt extractor and, fingers crossed, no metal shavings will drop in through the hole.
Maybe add the JB Weld in small steps, using a toothpick to spread it around the walls of the hole so there's never enough in one spot to run. Add more as the previous batch begins to dry.