Zack,
It sounds like at a minimum you would want 2 SymmetricDS installations. One for your company server and intermediate server. If these are on the same local network a multi-homed (local node) setup would work. I would make the company server the master node and the intermediate a local node in the same installation. The final installation of SymmetricDS would be close to your DB internet server (you want a reliable JDBC connection here). Then I would setup 3 node groups (ex: server, intermediate, web). I would build group links:
intermediate pulls from server (for your data)
intermediate pushes to server (for heartbeats and config)
intermediate pushes to web (for your data and allows you to keep a firewall in place from outside connections in).
intermediate pulls from web (for heartbeats and config)
From here you can select your tables for replication and should be all set.