Synchronizing two SQL Server database tables with the most recent changes to both tables
How often would you be writing to the master? And what purpose is Master serving in your setup?

Merge or peer-to-peer replication is one solution to keep both databases in sync but keep in mind they can be time consuming to troubleshoot and adds another extra thing in the mix to worry about.

Typically this would be done by updating records based on ID and last_updated key on tables. Merge statements can be a cause for race conditions where multiple users try and run the merge at the same time. Doing these operations with individual statements may keep you more clarity on the different operations that are happening on the db and keep things simple.

Delete: I would do this first by marking the records as 'soft-delete' (can be bit value) on master. Then running a sql server job on a schedule to remove all the records from tables which are marked for deletion. 

Insert: Pull new records down from master based on the primary key value being higher than what's in the client.

Update: Based on the last_update field. This will solve the issue of updating tables on the remote server as well.