Note for posterity: I updated DBD::SQLite to 1.46 and the problem went away. The unicode tokenizer works now.
It might be due to a change in SQLite itself; the changelog at http://www.sqlite.org/changes.html says: The unicode61 tokenizer is now included in FTS4 by default. (as of 3.8.6)
The previous DBD::SQLite version I used included v3.7.x of SQLite, the update got me up to 3.8.7.2.