What can a company that has been exposed as a sweatshop do to remain viable?
The best thing for a company to do when confronted with this sort of a crisis is to do a good job of communicating with the public about the scandal.
The company must react quickly. It must not look like it is stalling or trying to avoid the issue. Instead, it must quickly get its message out. It must get that message out in a coherent and consistent way. It must, however, make sure that this message is absolutely truthful. If you tell the truth (and then lay out a strategy for mending your ways in a timely fashion), the scandal might go away. However, if you lie and your are caught in that lie, your company will lose all credibility. The exposure of the lie will also keep the scandal in the public eye. This will not help you at all.
So the only way to survive this sort of thing is to respond quickly and honestly to the scandal. You must explain what happened, tell why (if there is any excuse) and then offer a realistic plan by which you will reform your practices.