You can use LiteSync with Perl. Instructions here:
But for this specific application you will need to check how to implement the changes. Maybe just using the modified SQLite driver and changing the db parameter may solve. Otherwise you will need to change the app code.
One server will be chosen as a primary node and it will use a config like this:
On the secondary node:
Note that with LiteSync we cannot have more than 1 app opening the same database file at the same time. Each app must open a local db file that will be synchronized to the others. So the above examples apply only if they are used on different computers. On the same computer we must use different file names or paths.