I believe we resolved the issue. After realizing that we had published two geodata services that referenced the same enterprise geodatabase version connection file, we removed one of the services from the ArcGIS Server site. Now when creating and editing a checkout replica, all of the edits are immediately synchronized to the SDE.Production version and available in the next round of replication. There is no synchronization delay that appeared to be occurring in the background.
Hopefully this helps anyone else that is having issues with ArcGIS Server and checkout replicas!