Can I apply production statistics to an empty local database to test PostgreSQL planner hypotheses? I'm toying with the idea of exporting a database's statistics from production and applying them to a test database (same structure, small dataset). The goal is to test software for changes that introduce expensive query plans in a non-production environment before the change goes to production.

I don't see any prior art on this. How might I go about applying statistics from one database instance to another instance, and lock those statistics down to perform this sort of test?
You cannot readily do that.  pg_statistic uses a bit of magic to be able to store the statistics in a column of type anyarray.  There is no ready way to access this magic from user-land, even if that user is the super-user.