This is very interesting. Establishing damages over reproduction of ones personal documents seems like it would be almost impossible to establish unless an actual crime had also taken place.

That's pretty much exactly right. However I think there is something to be said along the lines of "What reason would you have for copying the documents, if not to commit an offense?"

This train of thought for me seems to lead towards the most satisfying justifications I can think of for why media piracy is probably morally justifiable.

I feel like the main reason some businesses are completely against piracy is because it helps keep their prices in check. Many businesses take the piss with pricing, however when it gets bad enough people look to alternatives. If there is no alternative, if piracy wasn't an option, then businesses would get away with ripping off people even more than they currently do.