Sounds like something you can set up with Filemaker server (Don't see a need for server advanced unless you also need to publish to the web or use ODBC/JDBC) I do recommend one copy of Filemaker Pro Advanced for whoever will design and maintain your database. It's value added design tools will save a lot of time and avoid a lot of frustration when troubleshooting design issues in your database.
You might want to down load the free trials and see how well they work for you.
Take a look at GoZync http://www.seedcode.com/cp-app/ste_cat/gozync
The developers have worked out a lot of the issues already - including those that many of us might stumble into unexpectedly. Depending upon how your 'central office' works, it may even be that Pro will do the 'central file' job rather than necessarily going to Server. That said, if the budget can afford Server, get it! It's fabulous for backups in particular. Or, there's also hosting as an option.
Regards, Mardi Kennedy