The main problem with drop box an similar such approaches is that it is very easy for one person's changes to the file to overwrite those made by another when they close the file and it re-uploads back to drop box. The method you describe--which is intended to keep more than one user from accessing the DB at a time sounds like it will work--but due to the vagaries of the internet--the email may arrive after the second user also opens the file and starts editing it. Saving a back up copy to a unique file name or to a location outside the drop box folder would be absoluteyl essential then, as it will keep you from losing data and edits that way, but you are still left with the task of then merging changes from two files--a task that may be simple or nearly impossible depending on the design of the database and the type of changes made to the data.
You can both open it so you both have a copy of FM Client.
You are connected to Dropbox so you both have an internet connection.
So why not just share it, live? (A 2Mb file? Are you telling me that, in all your attic/basement/friends' basements/local garage sales/charity shop you cannot find an old XP workstation to connect to the web and run it on?)