[The Squatter of Amon Rûdh]

I've never seen or heard the phrase "x is the mischief" anywhere else either, and I'm pretty sure that a proof reader or Tolkien himself would have spotted this if it had been a mistake. I'm reasonably sure that he was using 'mischief' as a synonym of 'trouble' or 'problem', but it could also be a joke.

'Mischief' isn't a typical Tolkienian word, since it's of Norman French origin. Its original form in Old French was meschever, meaning roughly 'come to a bad end', and it bears no relationship with English 'chief', which derives from chief/chef. It sounds similar, however, and in a story for children it's a harmless enough joke to suggest that there can be a mis-chief, which is the foremost misfortune. In the circumstances, that's not an unreasonable description of the purse Bilbo tries to steal.