Sync two mysql database one in intranet and the other on internet
MySQL has replication cabapilities, sure you has revised the MySQL documentation.

This means that MySQL copies data from master to slave, not both directions.

You proposed "just click a button", if you don't want a "do it yourself" solution maybe can try with a database tool like MySQL workbench.

In any case it is dangerous to synchronize two databases with files: data inconsistencies, primary keys collisions, ...