What postgres sync solution should I use to duplicate production, but read it from another application?
It sounds like streaming replication with a hot standby is the solution you're looking for.

Streaming Replication details the steps of how you go about setting it up. The hot standby option that they go over (they have it put to "on" which is what you will want) will allow you to run SELECTS against the 2nd database. Under this setup, however, your production application would only write everything to the master database ... postgres itself would then stream those changes to the 2nd database.