Logic for getting prices is based on the Special Price attribute, but in case we have catalog price rule sale price in FB Catalog will not be populated. This is visible here.

As additional issue we never check is this sale price in the time frame or not.

Steps to reproduce (*)

1. Create new Catalog Price Rule - Discount everything with 10%
2. Save product or Push catalog manually

Expected result (*)

1. Products have sale price in the Catalog Manager

Actual result (*)

1. Products have only regular price.