IF you can find an old hardware store that might have Val-oil, snatch it up immediately, make a slurry with that and the cork dust, and build and fill with that!
Betting all of us old geezers here will tell you that THAT WAS THE IT way to go when working with black cork.
As far as a 3work preference, work outdoors, using a big tarp to catch what you grind or rasp. Easy way to save it.
Definitely use Rooster's method on joining the big pieces.. I almost suggested grinding that really beat one up to use as filler, BUT on second thought, you can always save with butt joints and some judiciously applied dowels for stability.
Looks like you have a fun time on your hands. Best to leave until after season, though. You might find this repair stuff becomes rather addictive!!!