Yeah, I would go the "shit looks bad, then full tilt" route. Have the party arrested, threatened with execution as soon as the judge files his report, etc. Then BAM, something terrible happens while they're in their cells, the whole ship is put in danger, and the judge and captain get toasted before they can file reports on what they saw. Maybe down the line another witness shows up who begins blackmailing them.
But I also like Ross' idea of using the built-in game mechanics as an immediate cost to the party. Systems that offer those kinds of mechanics usually play better if you turn to those mechanics first. I would personally put it up to the party, "OK, you guys know your characters are now in deep shit. I'll offer you an out if you burn all of your Fate points, or we can play through the terrible shit you brought down on yourselves."