As an optimization I just worked on for my database earlier this week, I decided to logically replicate that table from my main authentication database into a each cluster, and I replaced all references to the FDW for read-only queries to use the logically replicated table. All write queries still hit the FDW as before.
Perhaps you considered this, but if you had not wanted to deal with the administration side with replication, and the centralized data is changed infrequently from only one application/source perhaps, then updating the source and then refreshing a materialized view on each local db that pulls in the foreign data could be a good option. Some chance of stale data since the refresh must be triggered, but for some use cases it may be the simplest setup.