I only did expedition once (and I hate how it's too open-ended with few satisfying end knots), but I never lose any supplies in 30 hours-ish play, hell I don't even know what's "camp supplies" even mean, since there's no difference whatsoever on both camp's looks and shop before and after prof kidnap and giving few supplies to pirates.
As for raid gimmick, just cover the entire beach with bear traps and activate the laser thingy. No one will die even if you don't helped them